NCERT Solutions For Class 12 Maths Chapter 4 – Determinants Class 12 NCERT Solutions

NCERT Solutions For Class 12 Maths Chapter 4 will clear all the doubts which were related to the 4th chapter. Class 12 Maths NCERT Solutions will help in improving your problem-solving abilities. Students must practice the Class 12 Maths NCERT Book Solutions. We give topic-wise solutions for every question in Class 12 Maths NCERT Solutions and the Problems are solved by India’s best teachers. Students can prepare for CBSE Class 12 Maths NCERT Solutions Chapter 4 through this article for getting a better score in the CBSE exams. If the students practice the solutions multiple times, then it will be easier for them to solve the toughest problems that are appeared in the examinations. NCERT Solutions For Class 12 gives you the best results in CBSE board exams. You can clarify all doubts regarding this chapter 4 by simply downloading it.

Class 12 Maths NCERT Solutions Chapter 4 – Determinants

Name Of The Section Topic Name
4 Determinants
4.1 Introduction
4.2 Determinant
4.3 Properties of Determinants
4.4 Area of Triangle
4.5 Adjoint and Inverse of a Matrix
4.6 Summary

 

NCERT Solutions For Class 12 Maths Chapter 4 Determinants Ex 4.1 – Introduction

Question 1:
Evaluate the determinants in Exercises 1 and 2.
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 1
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 1
= 2(−1) − 4(−5) = − 2 + 20 = 18


Question 2:
Evaluate the determinants in Exercises 1 and 2.
(i) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 2(ii) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 2(a)
Answer:
(i) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 2 = (cos θ)(cos θ) − (−sin θ)(sin θ) = cos2 θ+ sin2 θ = 1
(ii) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 2(a)
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 2(b)

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Question 3:
If NCERT Solutions class 12 maths chapter 4 ex 4.1 q 3then show that NCERT Solutions class 12 maths chapter 4 ex 4.1 q 3(a)
Answer:
The given matrix is NCERT Solutions class 12 maths chapter 4 ex 4.1 q 3
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 3(b)


Question 4:
If NCERT Solutions class 12 maths chapter 4 ex 4.1 q 4 then show that NCERT Solutions class 12 maths chapter 4 ex 4.1 q 4(a)
Answer:
The given matrix is NCERT Solutions class 12 maths chapter 4 ex 4.1 q 4
It can be observed that in the first column, two entries are zero. Thus, we expand along the first column (C1) for easier calculation.
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 4(b)
From equations (i) and (ii), we have:
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 4(a)
Hence, the given result is proved.


Question 5:

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Evaluate the determinants
(i) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(ii) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(a)(iii) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(b)(iv) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(c)
Answer:
(i) Let A= NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5
It can be observed that in the second row, two entries are zero. Thus, we expand along the second row for easier calculation.
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(d)
(ii) Let A= NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(a)

By expanding along the first row, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(e)
(iii) Let A= NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(b)

By expanding along the first row, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(f)
(iv) Let A= NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(c)

By expanding along the first column, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 5(g)


Question 6:
If NCERT Solutions class 12 maths cahpter 4 ex 4.1 q 6find NCERT Solutions class 12 maths chapter 4 ex 4.1 q 6(a)
Answer:
Let NCERT Solutions class 12 maths cahpter 4 ex 4.1 q 6

By expanding along the first row, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 6(b)


Question 7:

Find values of x, if

(i) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 7(ii) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 7(a)

Answer:
(i) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 7
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 7(b)
(ii) NCERT Solutions class 12 maths chapter 4 ex 4.1 q 7(a)
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 7(c)


Question 8:
If NCERT Solutions class 12 maths chapter 4 ex 4.1 q 8

then x is equal to

(A) 6 (B) ±6 (C) −6 (D) 0
Answer:
Answer: B
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 8
NCERT Solutions class 12 maths chapter 4 ex 4.1 q 8(a)

Hence, the correct answer is B.

NCERT Solutions Class 12 Maths Chapter 4 Determinants Ex 4.2

Question 1:
Using the property of determinants and without expanding, prove that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 1
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 1(a)



Question 2:

Using the property of determinants and without expanding, prove that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 2
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 2(a)

Here, the two rows R1 and R3 are identical.

Δ = 0.


Question 3:

Using the property of determinants and without expanding, prove that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 3
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 3(a)



Question 4:

Using the property of determinants and without expanding, prove that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 4
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 4(a)

By applying C→ C3 + C2, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 4(b)

Here, two columns C1 and Care proportional.

Δ = 0.


Question 5:
Using the property of determinants and without expanding, prove that
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 5
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 5(a)

Applying R2 → R2 − R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 5(b)

Applying R1 ↔R3 and R2 ↔R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 5(c)

Applying R→ R1 − R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 5(d)

Applying R1 ↔R2 and R2 ↔R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 5(e)

From (1), (2), and (3), we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 5(f)

Hence, the given result is proved.


Question 6:

By using properties of determinants, show that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 6
Answer:
We have,
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 6(a)

Here, the two rows R1 and Rare identical.

∴Δ = 0.


Question 7:

By using properties of determinants, show that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 7
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 7(a)

Applying R→ R2 + R1 and R→ R3 + R1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 7(b)


Question 8:

By using properties of determinants, show that:
(i) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8
(ii) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8(a)
Answer:
(i) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8(b)

Applying R1 → R1 − Rand R2 → R2 − R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8(c)

Applying R1 → R1 + R2, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8(d)

Expanding along C1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8(e)

Hence, the given result is proved.
(ii) Let NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8(f)

Applying C1 → C1 − Cand C2 → C2 − C3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8(g)

Applying C1 → C1 + C2, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8(h)

Expanding along C1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 8(i)

Hence, the given result is proved.


Question 9:

By using properties of determinants, show that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 9
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 9(a)

Applying R2 → R2 − Rand R3 → R3 − R1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 9(b)

Applying R3 → R3 + R2, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 9(c)

Expanding along R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 9(d)

Hence, the given result is proved.


Question 10:

By using properties of determinants, show that:
(i) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10
(ii) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10(a)
Answer:
(i) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10(b)

Applying R1 → R1 + R+ R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10(c)

Applying C2 → C2 − C1, C3 → C3 − C1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10(d)

Expanding along C3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10(e)

Hence, the given result is proved.
(ii) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10(f)

Applying R1 → R1 + R+ R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10(g)

Applying C2 → C2 − Cand C3 → C3 − C1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10(h)

Expanding along C3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 10(i)

Hence, the given result is proved.


Question 11:

By using properties of determinants, show that:
(i) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11
(ii) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11(a)
Answer:
(i) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11(b)

Applying R1 → R1 + R+ R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11(c)

Applying C2 → C2 − C1, C3 → C3 − C1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11(d)

Expanding along C3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11(e)

Hence, the given result is proved.
(ii) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11(f)

Applying C1 → C1 + C+ C3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11(g)

Applying R2 → R2 − Rand R3 → R3 − R1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11(h)

Expanding along R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 11(i)

Hence, the given result is proved.


Question 12:

By using properties of determinants, show that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 12
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 12(a)

Applying R1 → R1 + R+ R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 12(b)

Applying C2 → C2 − Cand C3 → C3 − C1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 12(c)

Expanding along R1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 12(d)

Hence, the given result is proved.


Question 13:

By using properties of determinants, show that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 13
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 13(a)

Applying R1 → R1 + bRand R2 → R2 − aR3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 13(b)

Expanding along R1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 13(c)


Question 14:

By using properties of determinants, show that:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 14
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 14(a)

Taking out common factors ab, and c from R1, R2, and Rrespectively, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 14(b)

Applying R2 → R2 − Rand R3 → R3 − R1, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 14(c)

Applying C1 → aC1, C→ bC2, and C3 → cC3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 14(d)

Expanding along R3, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 14(e)

Hence, the given result is proved.


Question 15:

Choose the correct answer.
Let A be a square matrix of order 3 × 3, then NCERT Solutions class 12 maths chapter 4 ex 4.2 q 15is equal to
(A) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 15(a)(B) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 15(b)(C) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 15(c)(D) NCERT Solutions class 12 maths chapter 4 ex 4.2 q 15(d)
Answer:
Answer: C
A is a square matrix of order 3 × 3.
NCERT Solutions class 12 maths chapter 4 ex 4.2 q 15(e)

Hence, the correct answer is C.


Question 16:

Which of the following is correct?

A. Determinant is a square matrix.

B. Determinant is a number associated to a matrix.

C. Determinant is a number associated to a square matrix.

D. None of these
Answer:

Answer: C
We know that to every square matrix, NCERT Solutions class 12 maths chapter 4 ex 4.2 q 16of order n. We can associate a number called the determinant of square matrix A, where NCERT Solutions class 12 maths chapter 4 ex 4.2 q 16(a)

element of A.

Thus, the determinant is a number associated to a square matrix.

Hence, the correct answer is C.

NCERT Solutions For Class 12 Maths Chapter 4 Determinants Ex 4.3 Properties of Determinants

Question 1:

Find area of the triangle with vertices at the point given in each of the following:

(i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8)

(iii) (−2, −3), (3, 2), (−1, −8)
Answer:
(i) The area of the triangle with vertices (1, 0), (6, 0), (4, 3) is given by the relation,
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 1

(ii) The area of the triangle with vertices (2, 7), (1, 1), (10, 8) is given by the relation,
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 1(a)

(iii) The area of the triangle with vertices (−2, −3), (3, 2), (−1, −8)

is given by the relation,
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 1(b)
Hence, the area of the triangle is NCERT Solutions class 12 maths chapter 4 ex 4.3 q 1(c)


Question 2:

Show that points NCERT Solutions class 12 maths chapter 4 ex 4.3 q 2 are collinear
Answer:

Area of ΔABC is given by the relation,
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 2(a)

Thus, the area of the triangle formed by points A, B, and C is zero.

Hence, the points A, B, and C are collinear.


Question 3:

Find values of k if area of triangle is 4 square units and vertices are

(i) (k, 0), (4, 0), (0, 2) (ii) (−2, 0), (0, 4), (0, k)
Answer:

We know that the area of a triangle whose vertices are (x1y1), (x2y2), and

(x3y3) is the absolute value of the determinant (Δ), where
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 3

It is given that the area of triangle is 4 square units.

∴Δ = ± 4.

(i) The area of the triangle with vertices (k, 0), (4, 0), (0, 2) is given by the relation,
Δ =NCERT Solutions class 12 maths chapter 4 ex 4.3 q 3(a)
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 3(b)

k + 4 = ± 4

When −k + 4 = − 4, k = 8.

When −k + 4 = 4, k = 0.

Hence, k = 0, 8.

(ii) The area of the triangle with vertices (−2, 0), (0, 4), (0, k) is given by the relation,
Δ =NCERT Solutions class 12 maths chapter 4 ex 4.3 q 3(c)
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 3(d)

k − 4 = ± 4

When k − 4 = − 4, k = 0.

When k − 4 = 4, k = 8.

Hence, k = 0, 8.


Question 4:

(i) Find equation of line joining (1, 2) and (3, 6) using determinants

(ii) Find equation of line joining (3, 1) and (9, 3) using determinants
Answer:
(i) Let P (xy) be any point on the line joining points A (1, 2) and B (3, 6). Then, the points A, B, and P are collinear. Therefore, the area of triangle ABP will be zero.
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 4

Hence, the equation of the line joining the given points is y = 2x.

(ii) Let P (xy) be any point on the line joining points A (3, 1) and

B (9, 3). Then, the points A, B, and P are collinear. Therefore, the area of triangle ABP will be zero.
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 4(a)
Hence, the equation of the line joining the given points is x − 3y = 0.


Question 5:

If area of triangle is 35 square units with vertices (2, −6), (5, 4), and (k, 4). Then k is

A. 12 B. −2 C. −12, −2 D. 12, −2
Answer:

Answer: D

The area of the triangle with vertices (2, −6), (5, 4), and (k, 4) is given by the relation,
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 5

It is given that the area of the triangle is ±35.

Therefore, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.3 q 5(a)

When 5 − k = −7, k = 5 + 7 = 12.

When 5 − k = 7, k = 5 − 7 = −2.

Hence, k = 12, −2.

The correct answer is D.

NCERT Solutions For Class 12 Maths Chapter 4 Determinants Ex 4.4 Area Of Triangle

Question 1:

Write Minors and Cofactors of the elements of the following determinants:
(i) NCERT Solutions class 12 chapter 4 ex 4.4 q 1(ii) NCERT Solutions class 12 maths chapter 4 ex 4.4 q 1(a)
Answer:
(i) The given determinant is NCERT Solutions class 12 chapter 4 ex 4.4 q 1

Minor of element aij is Mij.

∴M11 = minor of element a11 = 3

M12 = minor of element a12 = 0

M21 = minor of element a21 = −4

M22 = minor of element a22 = 2

Cofactor of aij is Aij = (−1)i + j Mij.

∴A11 = (−1)1+1 M11 = (−1)2 (3) = 3

A12 = (−1)1+2 M12 = (−1)3 (0) = 0

A21 = (−1)2+1 M21 = (−1)3 (−4) = 4

A22 = (−1)2+2 M22 = (−1)4 (2) = 2
(ii) The given determinant is NCERT Solutions class 12 maths chapter 4 ex 4.4 q 1(a)

Minor of element aij is Mij.

∴M11 = minor of element a11 d

M12 = minor of element a12 b

M21 = minor of element a21 c

M22 = minor of element a22 a

Cofactor of aij is Aij = (−1)i + j Mij.

∴A11 = (−1)1+1 M11 = (−1)2 (d) = d

A12 = (−1)1+2 M12 = (−1)3 (b) = −b

A21 = (−1)2+1 M21 = (−1)3 (c) = −c

A22 = (−1)2+2 M22 = (−1)4 (a) = a



Question 2:
(i) NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(ii) NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(a)
Answer:
(i) The given determinant is NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2

By the definition of minors and cofactors, we have:
M11 = minor of a11=NCERT Solutions class 12 maths chapter 4 ex 4.4 2(b)

M12 = minor of a12NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(c)

M13 = minor of a13 = NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(d)

M21 = minor of a21 = NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(c)

M22 = minor of a22 =NCERT Solutions class 12 maths chapter 4 ex 4.4 2(b)

M23 = minor of a23 = NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(e)

M31 = minor of a31= NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(f)

M32 = minor of a32 = NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(e)

M32 = minor of a32 =NCERT Solutions class 12 maths chapter 4 ex 4.4 2(b)

A11 = cofactor of a11= (−1)1+1 M11 = 1

A12 = cofactor of a12 = (−1)1+2 M12 = 0

A13 = cofactor of a13 = (−1)1+3 M13 = 0

A21 = cofactor of a21 = (−1)2+1 M21 = 0

A22 = cofactor of a22 = (−1)2+2 M22 = 1

A23 = cofactor of a23 = (−1)2+3 M23 = 0

A31 = cofactor of a31 = (−1)3+1 M31 = 0

A32 = cofactor of a32 = (−1)3+2 M32 = 0

A33 = cofactor of a33 = (−1)3+3 M33 = 1
(ii) The given determinant is NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(a)

By definition of minors and cofactors, we have:
M11 = minor of a11= NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(g)

M12 = minor of a12=NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(h)

M13 = minor of a13 =NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(i)

M21 = minor of a21 = NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(j)

M22 = minor of a22 =NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(k)

M23 = minor of a23 = NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(l)

M31 = minor of a31= NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(m)

M32 = minor of a32 = NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(n)

M33 = minor of a33 = NCERT Solutions class 12 maths chapter 4 ex 4.4 q 2(o)

A11 = cofactor of a11= (−1)1+1 M11 = 11

A12 = cofactor of a12 = (−1)1+2 M12 = −6

A13 = cofactor of a13 = (−1)1+3 M13 = 3

A21 = cofactor of a21 = (−1)2+1 M21 = 4

A22 = cofactor of a22 = (−1)2+2 M22 = 2

A23 = cofactor of a23 = (−1)2+3 M23 = −1

A31 = cofactor of a31 = (−1)3+1 M31 = −20

A32 = cofactor of a32 = (−1)3+2 M32 = 13

A33 = cofactor of a33 = (−1)3+3 M33 = 5


Question 3:
Using Cofactors of elements of the second row, evaluate NCERT Solutions class 12 maths chapter 4 ex 4.4 q 3
Answer:
The given determinant is NCERT Solutions class 12 maths chapter 4 ex 4.4 q 3(a)

We have:
M21 = NCERT Solutions class 12 maths chapter 4 ex 4.4 q 3(b)
∴A21 = cofactor of a21 = (−1)2+1 M21 = 7
M22 =NCERT Solutions class 12 maths chapter 4 ex 4.4 q 3(c)

∴A22 = cofactor of a22 = (−1)2+2 M22 = 7
M23 =NCERT Solutions class 12 maths chapter 4 ex 4.4 q 3(d)

∴A23 = cofactor of a23 = (−1)2+3 M23 = −7

We know that Δ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.

∴Δ = a21A21 + a22A22 + a23A23 = 2(7) + 0(7) + 1(−7) = 14 − 7 =7


Question 4:
Using Cofactors of elements of third column, evaluate
NCERT Solutions class 12 maths cahpter 4 ex 4.4 q 4
Answer:
The given determinant is NCERT Solutions class 12 maths cahpter 4 ex 4.4 q 4(a)

We have:
M13 = NCERT Solutions class 12 maths cahpter 4 ex 4.4 q 4(b)
M23 = NCERT Solutions class 12 maths cahpter 4 ex 4.4 q 4(c)
M33 = NCERT Solutions class 12 maths cahpter 4 ex 4.4 q 4(d)

∴A13 = cofactor of a13 = (−1)1+3 M13 = (z − y)

A23 = cofactor of a23 = (−1)2+3 M23 = − (z − x) = (x − z)

A33 = cofactor of a33 = (−1)3+3 M33 = (y − x)

We know that Δ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.

NCERT Solutions class 12 maths chapter 4 ex 4.4 q 4(e)
Hence Δ=(x-y)(y-z)(z-x)


Question 5:
If NCERT Solutions class 12 maths chapter 4 ex 4.4 q 5and Aij is Cofactors of aij, then value of Δ is given by
NCERT Solutions class 12 maths chapter 4 ex 4.4 q 5(a)
Answer:

Answer: D

We know that:

Δ = Sum of the product of the elements of a column (or a row) with their corresponding cofactors

∴Δ = a11A11 + a21A21 + a31A31

Hence, the value of Δ is given by the expression given in alternative D.

The correct answer is D.

NCERT Solutions For Class 12 Maths Chapter 4 Determinants Ex 4.5 Adjoint and Inverse of a Matrix

Question 1:

Find adjoint of each of the matrices.
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 1
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 1(a)



Question 2:

Find adjoint of each of the matrices.
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 2
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 2(a)



Question 3:
Verify A (adj A) = (adj AA = NCERT Solutions class 12 maths chapter 4 ex 4.5 q 3
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 3(a)
Answer:
A= NCERT Solutions class 12 maths chapter 4 ex 4.5 q 3(a)
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 3(b)


Question 4:
Verify A (adj A) = (adj AA = NCERT Solutions class 12 maths chapter 4 ex 4.5 q 4
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 4(a)
Answer:
NCERT Solutions class 12 maths cahpter 4 ex 4.5 q 4(b)
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 4(b)


Question 5:

Find the inverse of each of the matrices (if it exists).
NCERT Solutions class 12 maths cahpter 4 ex 4.5 q 5
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 5(a)



Question 6:

Find the inverse of each of the matrices (if it exists).
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 6
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 6(a)



Question 7:

Find the inverse of each of the matrices (if it exists).
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 7
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 7(a)



Question 8:

Find the inverse of each of the matrices (if it exists).
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 8
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 8(a)



Question 9:

Find the inverse of each of the matrices (if it exists).
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 9
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 9(a)



Question 10:

Find the inverse of each of the matrices (if it exists).
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 10
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 10(a)



Question 11:

Find the inverse of each of the matrices (if it exists).
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 11
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 11(a)



Question 12:
Let NCERT Solutions class 12 maths chapter 4 ex 4.5 q 12and NCERT Solutions class 12 maths chapter 4 ex 4.5 q 12(a)Verify that NCERT Solutions class 12 maths chapter 4 ex 4.5 q 12(b)
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 12(c)
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 12(d)

 

From (1) and (2), we have:

(AB)−1 = B−1A−1

Hence, the given result is proved.


Question 13:
If NCERT Solutions class 12 maths chapter 4 ex 4.5 q 13show that NCERT Solutions class 12 maths chapter 4 ex 4.5 q 13(a)Hence find NCERT Solutions class 12 maths chapter 4 ex 4.5 q 13(b)
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 13(c)
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 13(d)



Question 14:
For the matrix NCERT Solutions class 12 maths chapter 4 ex 4.5 q 14find the numbers a and b such that A2 + aA + bI O.
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 14(a)
We have:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 14(b)Comparing the corresponding elements of the two matrices, we have:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 14(c)
Hence, −4 and 1 are the required values of a and b respectively.



Question 15:
For the matrix NCERT Solutions class 12 maths chapter 4 ex 4.5 q 15show that A3 − 6A2 + 5A + 11 I = O. Hence, find A−1.
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 15(a)
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 15(b)
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 15(c)

From equation (1), we have:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 15(d)


Question 16:
If NCERT Solutions class 12 maths chapter 4 ex 4.5 q 16verify that A3 − 6A2 + 9A − 4I = O and hence find A−1
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 16(a)
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 16(b)

From equation (1), we have:
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 16(c)


Question 17:
Let A be a nonsingular square matrix of order 3 × 3. Then NCERT Solutions class 12 maths chapter 4 ex 4.5 q 17is equal to
(A) NCERT Solutions class 12 maths chapter 4 ex 4.5 q 17(a)(B) NCERT Solutions class 12 maths chapter 4 ex 4.5 q 17(b)(C) NCERT Solutions class 12 maths chapter 4 ex 4.5 q 17(c)(D) NCERT Solutions class 12 maths chapter 4 ex 4.5 q 17(d)
Answer:

Answer: B

We know that
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 17(e)

Hence, the correct answer is B.


Question 18:
If A is an invertible matrix of order 2, then det (A−1) is equal to A. det (AB NCERT Solutions class 12 maths chapter 4 ex 4.5 q 18C. 1 D. 0
Answer:
Since A is an invertible matrix, NCERT Solutions class 12 maths chapter 4 ex 4.5 q 18(a)
NCERT Solutions class 12 maths chapter 4 ex 4.5 q 18(b)

Hence, the correct answer is B.

NCERT Solutions For Class 12 Maths Chapter 4 Determinants Ex 4.6 Summary

Question 1:

Examine the consistency of the system of equations.

+ 2= 2

2x + 3= 3
Answer:

The given system of equations is:

+ 2= 2

2x + 3= 3

The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q1

∴ A is non-singular.

Therefore, A−1 exists.

Hence, the given system of equations is consistent.


Question 2:

Examine the consistency of the system of equations.

2− y = 5

x + = 4
Answer:

The given system of equations is:

2− y = 5

x + = 4

The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 2

∴ A is non-singular.

Therefore, A−1 exists.

Hence, the given system of equations is consistent.


Question 3:

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8
Answer:

The given system of equations is:

x + 3y = 5

2x + 6y = 8

The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 3

∴ A is a singular matrix.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 3(a)
Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.


Question 4:

Examine the consistency of the system of equations.

x + y z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4
Answer:

The given system of equations is:

x + y z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

This system of equations can be written in the form AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 4
∴ A is non-singular.

Therefore, A−1 exists.

Hence, the given system of equations is consistent.


Question 5:

Examine the consistency of the system of equations.

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3
Answer:

The given system of equations is:

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3

This system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 5

∴ A is a singular matrix.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 5(a)

Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.


Question 6:

Examine the consistency of the system of equations.

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1
Answer:

The given system of equations is:

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

This system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 6

∴ A is non-singular.

Therefore, A−1 exists.

Hence, the given system of equations is consistent.


Question 7:

Solve system of linear equations, using matrix method.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 7
Answer:
The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 7(a)

Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 7(b)


Question 8:

Solve system of linear equations, using matrix method.
2x-y=-2
3x+4y=3
Answer:
The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 8

Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 8(a)


Question 9:

Solve system of linear equations, using matrix method.
4x-3y=3
3x-5y=7
Answer:

The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 9

Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 9(a)


Question 10:

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5
Answer:
The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 10

Thus, A is non-singular. Therefore, its inverse exists.


Question 11:

Solve system of linear equations, using matrix method.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 11
Answer:

The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 11(a)

Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 11(b)


Question 12:

Solve system of linear equations, using matrix method.

x − y + z = 4

2x + y − 3z = 0

x + y + z = 2
Answer:

The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 12

Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 12(b)


Question 13:

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3x − y − 2z = 3
Answer:

The given system of equations can be written in the form AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 13

Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 13(a)


Question 14:

Solve system of linear equations, using matrix method.

x − y + 2z = 7

3x + 4y − 5z = −5

2x − y + 3z = 12
Answer:
The given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 14

Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 14(a)


Question 15:
If NCERT Solutions class 12 maths chapter 4 ex 4.6 q 15find A−1. Using A−1 solve the system of equations
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 15(a)
Answer:
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 15(b)

Now, the given system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 15(c)


Question 16:

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg

wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70.

Find cost of each item per kg by matrix method.
Answer:

Let the cost of onions, wheat, and rice per kg be Rs x, Rs y,and Rs z respectively.

Then, the given situation can be represented by a system of equations as
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 16

This system of equations can be written in the form of AX = B, where
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 16(a)

Now,

X = A−1 B
NCERT Solutions class 12 maths chapter 4 ex 4.6 q 16(b)

Hence, the cost of onions is Rs 5 per kg, the cost of wheat is Rs 8 per kg, and the cost of rice is Rs 8 per kg.


NCERT Solutions For Class 12 Maths Chapter 4 Determinants Miscellaneous Solutions

Question 1:
Prove that the determinant NCERT Solutions class 12 maths chapter 4 ms q 1is independent of θ.
Answer:
Δ=NCERT Solutions class 12 maths chapter 4 ms q 1
NCERT Solutions class 12 matsh chapter 4 ms q 1(a)


Question 2:

Without expanding the determinant, prove that
NCERT Solutions class 12 maths chapter 4 ms q 2
Answer:
NCERT Solutions class 12 matsh chapter 4 ms q 2(a)

Hence, the given result is proved.


Question 3:
Evaluate NCERT Solutions class 12 maths chapter 4 ms q 3
Answer:
Δ=NCERT Solutions class 12 maths chapter 4 ms q 3

Expanding along C3, we have:
NCERT Solutions class 12 maths chapter 4 ms q 3(a)


Question 4:
If ab and are real numbers, and NCERT Solutions class 12 matsh chapter 4 ms q 4
Show that either a + b + c = 0 or a = b = c.
Answer:
NCRET Solutions class 12 maths chapter 4 ms q 4

Expanding along R1, we have:
NCERT Solutions class 12 maths chapter 4 ms q 4(b)
NCERT Solutions class 12 maths chapter 4 ms q 4(c)

Hence, if Δ = 0, then either a + b + c = 0 or a = b = c.


Question 5:
Solve the equations NCERT Solutions class 12 maths chapter 4 ms q 5
Answer:
NCERT Solutions class 12 maths chapter 4 ms q 5(a)



Question 6:
Prove that NCERT Solutions class 12 maths chapter 4 ms q 6
Answer:
NCERT Solutions class 12 maths chapter 4 ms q 6(a)

Expanding along R3, we have:
NCERT Solutions class 12 maths chapter 4 ms q 6(b)

Hence, the given result is proved.


Question 7:
If NCERT Solutions class 12 maths chapter 4 ms q 7
Answer:
We know that NCERT Solutions class 12 maths chapter 4 ms q 7(a)
NCERT Solutions class 12 maths chapter 4 ms q 7(b)


Question 8:
Let NCERT Solutions class 12 maths chapter 4 ms q 8verify that
(i) NCERT Solutions class 12 maths chapter 4 ms q 8(a)
(ii) NCERT Solutions class 12 maths chapter 4 ms q 8(b)
Answer:
NCERT Solutions class 12 maths chapter 4 ms q 8(c)
(i)
NCERT Solutions class 12 maths chapter 4 ms q 8(d)

We have,
NCERT Solutions class 12 maths chapter 4 ms q 8(e)
(ii)
NCERT Solutions class 12 maths chapter 4 ms q 8(f)


Question 9:
Evaluate NCERT Solutions class 12 maths chapter 4 ms q 9
Answer:
Δ=NCERT Solutions class 12 maths chapter 4 ms q 9
NCERT Solutions class 12 maths chapter 4 ms q 9(a)

Expanding along R1, we have:
NCERT Solutions class 12 maths chapter 4 ms q 9(b)


Question 10:
Evaluate NCERT Solutions class 12 maths chapter 4 ms q 10
Answer:
NCERT Solutions class 12 maths chapter 4 ms q 10(a)

Expanding along C1, we have:
Δ=1(xy-0)=xy


Question 11:

Using properties of determinants, prove that:
NCERT Solutions class 12 maths chapter 4 ms q 11
Answer:
NCERT Solutions class 12 maths chapter 4 ms q 11(a)
Expanding along R3, we have:
NCERT Solutions class 12 maths chapter 4 ms q 11(b)

Hence, the given result is proved.


Question 12:

Using properties of determinants, prove that:

Answer:
NCERT Solutions class 12 maths chapter 4 ms q 12(a)

 

Expanding along R3, we have:
NCERT Solutions class 12 maths chapter 4 ms q 12(b)

Hence, the given result is proved.


Question 13:

Using properties of determinants, prove that:
NCERT Solutions class 12 maths chapter 4 ms q 13
Answer:
NCERT Solutions class 12 maths chapter 4 ms q 13(a)

Expanding along C1, we have:
NCERT Solutions class 12 maths chapter 4 ms q 13(b)

Hence, the given result is proved.


Question 14:

Using properties of determinants, prove that:
NCERT Solutions class 12 maths chapter 4 ms q 14
Answer:
NCERT Solutions class 12 maths chapter 4 ms q 14(a)

Expanding along C1, we have:
NCERT Solutions class 12 maths chapter 4 ms q 14(b)

Hence, the given result is proved.


Question 15:

Using properties of determinants, prove that:
NCERT Solutions class 12 maths chapter 4 ms q 15
Answer:
NCERT Solutions class 12 maths chapter 4 ms q 15(a)

Hence, the given result is proved.


Question 16:

Solve the system of the following equations
NCERT Solutions class 12 maths chapter 4 ms q 16
Answer:
Let NCERT Solutions class 12 maths chapter 4 ms q 16(a)

Then the given system of equations is as follows:
NCERT Solutions class 12 maths chapter 4 ms q 16(b)

This system can be written in the form of AX B, where
NCERT Solutions class 12 maths chapter 4 ms q 16(c)

Thus, A is non-singular. Therefore, its inverse exists.

Now,

A11 = 75, A12 = 110, A13 = 72

A21 = 150, A22 = −100, A23 = 0

A31 = 75, A32 = 30, A33 = − 24
NCERT Solutions class 12 maths chapter 4 ms q 16(d)


Question 17:

Choose the correct answer.

If abc, are in A.P., then the determinant
NCERT Solutions class 12 maths chapter 4 ms q 17

A. 0 B. 1 C. D. 2x
Answer:

Answer: A
NCERT Solutions class 12 maths chapter 4 ms q 17(a)

Here, all the elements of the first row (R1) are zero.

Hence, we have Δ = 0.

The correct answer is A.


Question 18:

Choose the correct answer.

If xyz are nonzero real numbers, then the inverse of matrix
NCERT Solutions class 12 maths chapter 4 ms q 18is
(A) NCERT Solutions class 12 maths chapter 4 ms q 18(a)(B) NCERT Solutions class 12 maths chapter 4 ms q 18(b)(C) NCERT Solutions class 12 maths chapter 4 ms q 18(c)(D) NCERT Solutions class 12 maths chapter 4 ms q 18(d)
Answer:
Answer: A
NCERT Solutions class 12 maths chapter 4 ms q 18(e)

The correct answer is A.


Question 19:

Choose the correct answer.
Let NCERT Solutions class 12 maths chapter 4 ms q 19where 0 ≤ θ≤ 2π, then

A. Det (A) = 0

B. Det (A) ∈ (2, ∞)

C. Det (A) ∈ (2, 4)

D. Det (A)∈ [2, 4] Answer:

Answer: D
NCERT Solutions class 12 maths chapter 4 ms q 19(a)
Now, 0≤θ≤2π
NCERT Solutions class 12 maths chapter 4 ms q 19(b)
⇒-1≤sinθ≤1
The correct answer is D.