NCERT Solutions For Class 12 Maths Chapter 4 Determinants Ex 4.5

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Textbook	NCERT
Board	CBSE
Category	NCERT Solutions
Class	Class 12
Subject	Maths
Chapter	Chapter 4
Exercise	Class 12 Maths Chapter 4 Determinants Exercise 4.5
Number of Questions Solved	18

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.5

NCERT TEXTBOOK EXERCISES

Ex 4.5 Class 12 Maths Question 1.
$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}=A(say)$
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 = $\begin{bmatrix} 4 & -3 \ -2 & 1 \end{bmatrix}$
= $\begin{bmatrix} 4 & -3 \ -2 & 1 \end{bmatrix}$

Ex 4.5 Class 12 Maths Question 2.
$\left[ \begin{matrix} 1 & -1 & 2 \ 2 & 3 & 5 \ -2 & 0 & 1 \end{matrix} \right] =A(say)$

Solution:

${ A }{ 11 }={ (-1) }^{ 1+1 }M{ 11 }=\begin{vmatrix} 3 & 5 \ 0 & 1 \end{vmatrix}=3$
Similarly,

Verify A (adjA) = (adjA) •A = |A| I in Qs. 3 and 4.
Ex 4.5 Class 12 Maths Question 3.
$\begin{bmatrix} 2 & 3 \ -4 & 6 \end{bmatrix}=A(say)$

Solution:

|A| = 24

Ex 4.5 Class 12 Maths Question 4.
$\left[ \begin{matrix} 1 & -1 & 2 \ 3 & 0 & -2 \ 1 & 0 & 3 \end{matrix} \right] =A(say)$

Solution:

A₁₁ = 0, A₁₂ = – 11, A₁₃ = 0,
A₂₁ = – 3, A₂₂ = 1, A₂₃ = 1, A₃₁ = – 2
A₃₂ = 8, A₃₃ = – 3

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.
$\begin{bmatrix} 2 & -2 \ 4 & 3 \end{bmatrix}=A(say)$

Solution:

$\left| A \right| =\begin{vmatrix} 2 & -2 \ 4 & 3 \end{vmatrix}=6+8=14\neq 0$
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

Ex 4.5 Class 12 Maths Question 6.
$\begin{bmatrix} -1 & 5 \ -3 & 2 \end{bmatrix}=A(say)$

Solution:

$\left| A \right| =\begin{vmatrix} -1 & 5 \ -3 & 2 \end{vmatrix}=-2+15=13\neq 0$
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

Ex 4.5 Class 12 Maths Question 7.
$\left[ \begin{matrix} 1 & 2 & 3 \ 0 & 2 & 4 \ 0 & 0 & 5 \end{matrix} \right] =A$

Solution:

|A| = 10
$\left[ \begin{matrix} 1 & 2 & 3 \ 0 & 2 & 4 \ 0 & 0 & 5 \end{matrix} \right] =A$

Ex 4.5 Class 12 Maths Question 8.
$\left[ \begin{matrix} 1 & 0 & 0 \ 3 & 3 & 0 \ 5 & 2 & -1 \end{matrix} \right] =A$

Solution:

$\left| A \right| =\left| \begin{matrix} 1 & 0 & 0 \ 3 & 3 & 0 \ 5 & 2 & -1 \end{matrix} \right| =1\left| \begin{matrix} 3 & 0 \ 2 & -1 \end{matrix} \right| =-3\neq 0$
So, A is a non-singular matrix and therefore it is invertible. Let c_ij be cofactor of a_ij in A. Then, the cofactors of elements of A are given by

Ex 4.5 Class 12 Maths Question 9.
$\left[ \begin{matrix} 2 & 1 & 3 \ 4 & -1 & 0 \ -7 & 2 & 1 \end{matrix} \right] =A$

Solution:

|A| = $\left[ \begin{matrix} 2 & 1 & 3 \ 4 & -1 & 0 \ -7 & 2 & 1 \end{matrix} \right] =A$
= 2(-1-0)-1(4-0)+3(8-3)
So, A is non-singular matrix and therefore, it is invertible.

Ex 4.5 Class 12 Maths Question 10.
$\left[ \begin{matrix} 1 & -1 & 2 \ 0 & 2 & -3 \ 3 & -2 & 4 \end{matrix} \right] =A$

Solution:

|A| = $\left[ \begin{matrix} 1 & -1 & 2 \ 0 & 2 & -3 \ 3 & -2 & 4 \end{matrix} \right] =A$
= 1(8-6)+1(0+9)+2(0-6)
= 2+9-12
= -1≠0
∴A is invertible and
${ A }^{ -1 }=\frac { Adj\quad A }{ |A| }$

Ex 4.5 Class 12 Maths Question 11.
$\left[ \begin{matrix} 1 & 0 & 0 \ 0 & cos\alpha & sin\alpha \ 0 & sin\alpha & -cos\alpha \end{matrix} \right]$

Solution:

A = $\left[ \begin{matrix} 1 & 0 & 0 \ 0 & cos\alpha & sin\alpha \ 0 & sin\alpha & -cos\alpha \end{matrix} \right]$
adj A = $\left[ \begin{matrix} -1 & 0 & 0 \ 0 & -cos\alpha & -sin\alpha \ 0 & -sin\alpha & cos\alpha \end{matrix} \right]$
First find |A| = -cos²α-sin²α
=-1≠0

Ex 4.5 Class 12 Maths Question 12.
Let $A=\begin{bmatrix} 3 & 7 \ 2 & 5 \end{bmatrix},B=\begin{bmatrix} 6 & 8 \ 7 & 9 \end{bmatrix}$, verify that (AB)^-1 = B^-1A^-1

Solution:

Here |A| = $A=\begin{bmatrix} 3 & 7 \ 2 & 5 \end{bmatrix}$
= 15-14
= 1≠0
$Adj A=\begin{bmatrix} 5 & -7 \ -2 & 3 \end{bmatrix}$

Ex 4.5 Class 12 Maths Question 13.
If $A=\begin{bmatrix} 3 & 1 \ -1 & 2 \end{bmatrix}$ show that A² – 5A + 7I = 0,hence find A^-1

Solution:

$A=\begin{bmatrix} 3 & 1 \ -1 & 2 \end{bmatrix}$
$\begin{bmatrix} 8 & 5 \ -5 & 3 \end{bmatrix}$

Ex 4.5 Class 12 Maths Question 14.
For the matrix A = $\begin{bmatrix} 3 & 2 \ 1 & 1 \end{bmatrix}$ find file numbers a and b such that A²+aA+bI²=0. Hence, find A^-1.

Solution:

A = $\begin{bmatrix} 3 & 2 \ 1 & 1 \end{bmatrix}$
A²+aA+bI²=0

Ex 4.5 Class 12 Maths Question 15.
For the matrix $A=\left[ \begin{matrix} 1 & 1 & 1 \ 1 & 2 & -3 \ 2 & -1 & 3 \end{matrix} \right]$ Show that A³-6A²+5A+11I₃=0.Hence find A^-1

Solution:

A² = $\left[ \begin{matrix} 4 & 2 & 1 \ -3 & 8 & -14 \ 7 & -3 & 14 \end{matrix} \right]$

Ex 4.5 Class 12 Maths Question 16.
If $A=\left[ \begin{matrix} 2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2 \end{matrix} \right]$ show that A³-6A²+9A-4I=0 and hence, find A^-1

Solution:

We have $A=\left[ \begin{matrix} 2 & -1 & 1 \ -1 & 2 & -1 \ 1 & -1 & 2 \end{matrix} \right]$

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 = $\left[ \begin{matrix} { a }{ 11 } & { a }{ 12 } & { a }{ 13 } \ { a }{ 21 } & { a }{ 22 } & { a }{ 23 } \ { a }{ 31 } & { a }{ 32 } & { a }_{ 33 } \end{matrix} \right]$

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) $\ \frac { 1 }{ det.(A) }$
(c) 1
(d) 0

Solution:

|A|≠0
=> A^-1 exists => AA^-1 = I
|AA^-1| = |I| = I
=> |A||A^-1| = I
$|{ A }^{ -1 }|=\frac { 1 }{ |A| }$
Hence option (b) is correct.

NCERT Solutions for Class 12 Maths Chapter 4 Determinants Exercise 4.5 PDF

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Other Chapter of Class 12 Maths Chapter 4 Determinants

• Class 12 Maths Chapter 4 Determinants Ex 4.1
• Class 12 Maths Chapter 4 Determinants Ex 4.2
• Class 12 Maths Chapter 4 Determinants Ex 4.3
• Class 12 Maths Chapter 4 Determinants Ex 4.4
• Class 12 Maths Chapter 4 Determinants Ex 4.5
• Class 12 Maths Chapter 4 Determinants Ex 4.6