NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5

Get Free NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Determinants Exercise 4.5 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers. Detailed answers of all the questions in Chapter 4 Class 12 Determinants Ex 4.5 provided in NCERT Textbook.

Free download NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5 PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

The topics and sub-topics included in the Determinants chapter are the following:

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

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5

Find the adjoint of each of the matrices in Questions 1 and 2.

Ex 4.5 Class 12 Maths Question 1.
Image may be NSFW.
Clik here to view. $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}=A(say)$
Solution:
Let C_ij be cofactor of a_ij in A. Then, the cofactors of elements of A are given by
C₁₁ = (-1)¹⁺¹ (4) = 4; C₁₂ = (-1)¹⁺²(3) = -3
C₂₁ = (-1)²⁺¹ (2)= – 2; C₂₂ = (-1)²⁺² (1) = -1
Adj A = Image may be NSFW.
Clik here to view. $\begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix}$
= Image may be NSFW.
Clik here to view. $\begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix}$

Ex 4.5 Class 12 Maths Question 2.
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Clik here to view. $\left[ \begin{matrix} 1 & -1 & 2 \\ 2 & 3 & 5 \\ -2 & 0 & 1 \end{matrix} \right] =A(say)$
Solution:
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Clik here to view. ${ A }_{ 11 }={ (-1) }^{ 1+1 }M_{ 11 }=\begin{vmatrix} 3 & 5 \\ 0 & 1 \end{vmatrix}=3$
Similarly,
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 2

Verify A (adjA) = (adjA) •A = |A| I in Qs. 3 and 4.
Ex 4.5 Class 12 Maths Question 3.
Image may be NSFW.
Clik here to view. $\begin{bmatrix} 2 & 3 \\ -4 & 6 \end{bmatrix}=A(say)$
Solution:
|A| = 24
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 3

Ex 4.5 Class 12 Maths Question 4.
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Clik here to view. $\left[ \begin{matrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{matrix} \right] =A(say)$
Solution:
A₁₁ = 0, A₁₂= – 11, A₁₃= 0,
A₂₁= – 3, A₂₂= 1, A₂₃= 1, A₃₁ = – 2
A₃₂= 8, A₃₃= – 3
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 4

Find the inverse of each of the matrices (if it exists) given in Questions 5 to 11:

Ex 4.5 Class 12 Maths Question 5.
Image may be NSFW.
Clik here to view. $\begin{bmatrix} 2 & -2 \\ 4 & 3 \end{bmatrix}=A(say)$
Solution:
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Clik here to view. $\left| A \right| =\begin{vmatrix} 2 & -2 \\ 4 & 3 \end{vmatrix}=6+8=14\neq 0$
So, A is a non-singular matrix and therefore it is invertible. Let c_ij be cofactor of a_ij in A. Then, the cofactors of elements of A are given by
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Clik here to view. vedantu class 12 maths Chapter 4 Determinants 5

Ex 4.5 Class 12 Maths Question 6.
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Clik here to view. $\begin{bmatrix} -1 & 5 \\ -3 & 2 \end{bmatrix}=A(say)$
Solution:
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Clik here to view. $\left| A \right| =\begin{vmatrix} -1 & 5 \\ -3 & 2 \end{vmatrix}=-2+15=13\neq 0$
So, A is a non-singular matrix and therefore it is invertible. Let c_ij be cofactor of a_ij in A. Then, the cofactors of elements of A are given by
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 6

Ex 4.5 Class 12 Maths Question 7.
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Clik here to view. $\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{matrix} \right] =A$
Solution:
|A| = 10
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Clik here to view. $\left[ \begin{matrix} 1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5 \end{matrix} \right] =A$
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 7

Ex 4.5 Class 12 Maths Question 8.
Image may be NSFW.
Clik here to view. $\left[ \begin{matrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{matrix} \right] =A$
Solution:
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Clik here to view. $\left| A \right| =\left| \begin{matrix} 1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1 \end{matrix} \right| =1\left| \begin{matrix} 3 & 0 \\ 2 & -1 \end{matrix} \right| =-3\neq 0$
So, A is a non-singular matrix and therefore it is invertible. Let c_ij be cofactor of a_ijin A. Then, the cofactors of elements of A are given by
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 8

Ex 4.5 Class 12 Maths Question 9.
Image may be NSFW.
Clik here to view. $\left[ \begin{matrix} 2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1 \end{matrix} \right] =A$
Solution:
|A| = Image may be NSFW.
Clik here to view. $\left[ \begin{matrix} 2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1 \end{matrix} \right] =A$
= 2(-1-0)-1(4-0)+3(8-3)
So, A is non-singular matrix and therefore, it is invertible.
Image may be NSFW.
Clik here to view. vedantu class 12 maths Chapter 4 Determinants 9

Ex 4.5 Class 12 Maths Question 10.
Image may be NSFW.
Clik here to view. $\left[ \begin{matrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{matrix} \right] =A$
Solution:
|A| = Image may be NSFW.
Clik here to view. $\left[ \begin{matrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{matrix} \right] =A$
= 1(8-6)+1(0+9)+2(0-6)
= 2+9-12
= -1≠0
∴A is invertible and
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Clik here to view. ${ A }^{ -1 }=\frac { Adj\quad A }{ |A| }$
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 10

Ex 4.5 Class 12 Maths Question 11.
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Clik here to view. $\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & cos\alpha & sin\alpha \\ 0 & sin\alpha & -cos\alpha \end{matrix} \right]$
Solution:
A = Image may be NSFW.
Clik here to view. $\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & cos\alpha & sin\alpha \\ 0 & sin\alpha & -cos\alpha \end{matrix} \right]$
adj A = Image may be NSFW.
Clik here to view. $\left[ \begin{matrix} -1 & 0 & 0 \\ 0 & -cos\alpha & -sin\alpha \\ 0 & -sin\alpha & cos\alpha \end{matrix} \right]$
First find |A| = -cos²α-sin²α
=-1≠0
Image may be NSFW.
Clik here to view. vedantu class 12 maths Chapter 4 Determinants 11

Ex 4.5 Class 12 Maths Question 12.
Let Image may be NSFW.
Clik here to view. $A=\begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix},B=\begin{bmatrix} 6 & 8 \\ 7 & 9 \end{bmatrix}$ , verify that (AB)^-1 = B^-1A^-1
Solution:
Here |A| = Image may be NSFW.
Clik here to view. $A=\begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix}$
= 15-14
= 1≠0
Image may be NSFW.
Clik here to view. $Adj A=\begin{bmatrix} 5 & -7 \\ -2 & 3 \end{bmatrix}$
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 12
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Clik here to view.

Ex 4.5 Class 12 Maths Question 13.
If Image may be NSFW.
Clik here to view. $A=\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ show that A² – 5A + 7I = 0,hence find A^-1
Solution:
A = Image may be NSFW.
Clik here to view. $\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$
A² = Image may be NSFW.
Clik here to view. $\begin{bmatrix} 8 & 5 \\ -5 & 3 \end{bmatrix}$
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 13

Ex 4.5 Class 12 Maths Question 14.
For the matrix A = Image may be NSFW.
Clik here to view. $\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$ find file numbers a and b such that A²+aA+bI²=0. Hence, find A^-1.
Solution:
A = Image may be NSFW.
Clik here to view. $\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$
A²+aA+bI²=0
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 14

Ex 4.5 Class 12 Maths Question 15.
For the matrix Image may be NSFW.
Clik here to view. $A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \end{matrix} \right]$ Show that A³-6A²+5A+11I₃=0.Hence find A^-1
Solution:
A² = Image may be NSFW.
Clik here to view. $\left[ \begin{matrix} 4 & 2 & 1 \\ -3 & 8 & -14 \\ 7 & -3 & 14 \end{matrix} \right]$
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 15
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Clik here to view. vedantu class 12 maths Chapter 4 Determinants 15.1

Ex 4.5 Class 12 Maths Question 16.
If Image may be NSFW.
Clik here to view. $A=\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right]$ show that A³-6A²+9A-4I=0 and hence, find A^-1
Solution:
We have
Image may be NSFW.
Clik here to view. $A=\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right]$
Image may be NSFW.
Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 16
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Clik here to view.

Ex 4.5 Class 12 Maths Question 17.
Let A be a non-singular square matrix of order 3×3. Then | Adj A | is equal to:
(a) | A |
(b) | A |²
(c) | A |³
(d) 3 | A |
Solution:
Let A = Image may be NSFW.
Clik here to view. $\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right]$
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Determinants 17
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Clik here to view.
Dividing by | A |, |Adj. A| = | A |²
Hence, Part (b) is the correct answer.

Ex 4.5 Class 12 Maths Question 18.
If A is an invertible matrix of order 2, then det. (A^-1) is equal to:
(a) det. (A)
(b) Image may be NSFW.
Clik here to view. $\\ \frac { 1 }{ det.(A) }$
(c) 1
(d) 0
Solution:
|A|≠0
=> A^-1 exists => AA^-1 = I
|AA^-1| = |I| = I
=> |A||A^-1| = I
Image may be NSFW.
Clik here to view. $|{ A }^{ -1 }|=\frac { 1 }{ |A| }$
Hence option (b) is correct.

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Hindi Medium Ex 4.5

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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.5 Determinants
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.5 in Hindi Medium
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Clik here to view. 12 Maths exercise 4.5 in Hindi Medium
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Clik here to view. NCERT Solutions for Class 12 Maths Chapter 4 Exercise 4.5 Determinants
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