RD Sharma Solutions Class 12 Mathematics Chapter 6 VSA

Schools
RD Sharma Solutions
RD Sharma Solutions Class 12 Mathematics Chapter 6 VSA

RD Sharma Solutions Class 12 Mathematics Chapter 6 VSA

Kuldeep MauryaUpdated on 20 Jan 2022, 02:31 PM IST

RD Sharma provides the best material for maths for CBSE students. They are exam-oriented, extremely detailed, and contain many solved examples. They are preferred even more than NCERT materials as they include more concepts. A majority of CBSE schools in the country use RD Sharma books for maths. This means that the solutions that students will learn from the material might also appear in their exams.

Also Read - RD Sharma Solution for Class 9 to 12 Maths

RD Sharma Class 12 Solutions Chapter6 VSA Adjoint & Inverse of Matrix - Other Exercise

Adjoint and inverse of a Matrix Exercise very short answer type Question 1

Answer :
$adj A=\left[\begin{array}{rr} -2 & -4 \\ -7 & -3 \end{array}\right]$
Given:
$A=\left[\begin{array}{rr} -3 & 4 \\ 7 & -2 \end{array}\right]$
Hint ; Find the cofactors of the matrix A and then write adj A
Solution ;
$A=\left[\begin{array}{rr} -3 & 4 \\ 7 & -2 \end{array}\right]$
$c_{11}=-2$ $c_{12}=-4$ $c_{21}=-7$ $c_{22}=-3$
$adj A=\left[\begin{array}{rr} -2 & -4 \\ -7 & -3 \end{array}\right]$

Adjoint and inverse of a Matrix Exercise very short answer type Question 2

$Answer: 5$
$Given : A \text{ is a square matrix and }A($ adj $A)=5 I$
$Hint : \text{you must know the formula of }$A($ adj $A)$ \text{ to find the answer}$
$Solution:$
$A(a d j A)=5 I-----(1)$
We know that
$A(\operatorname{adj} A)=|A| I-------(2)$
$\text{on comparing (1) and }(2)$
$|A| I=5 I$
$|A|=5$

Adjoint and inverse of a Matrix Exercise very short answer type Question 3

Answer : 25
Given : A is a square matrix of order 3 and $\left | A \right |=5$
Hint : use the formula of $adj\left ( A \right )$
solution :
$n=3$ & $\left | A \right |=5$
We know that
$\begin{aligned} &|a d j A|=|A|^{n-1} \\ &=(5)^{3-1} \quad(n=3 \&|A|=5) \\ &=5^{2} \\ &|a d j A|=25 \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 5

Answer : $\frac{1}{10}$
Given : A is non-singular square matrix and $\left | A \right |=10$
you must know about the concept of non-singular matrix
Solution : We know that
$\left | A \right |^{-1}=\frac{1}{\left | A \right |}$
$\left | A \right |^{-1}=\frac{1}{10} \ \ \ \ \ \ \ \because \left ( \left | A \right |=10 \right)$

Adjoint and inverse of a Matrix Exercise very short answer type Question 6

Answer: A must be invertible and $\left | A \right |\neq 0$

Given: A, B and C are non null square matrix of same order & AB=AC implies B = C

Hint : you must know about invertible and non-invertible matrix

Solution : AB=AC can be write only when A is invertible$\left | A \right |\neq 0$

$|A| \neq 0$

$A B=A C$

Multiply both sides by $A^{-1}$

$A^{-1} A B=A^{-1} A C$

$I B=I C$

$B=C$

If Ais not invertible then $A^{-1}$ does `nt exist then,

$AB=AC$

$B=C$

Hence, the condition of A is invertible or $\left | A \right |\neq 0$

Adjoint and inverse of a Matrix Exercise very short answer type Question 7

Answer: $\left(A^{T}\right)^{-1}=\left[\begin{array}{ll} 5 & -2 \\ 3 & -1 \end{array}\right]$

Given: A is non singular square matrix and $\left(A\right)^{-1}=\left[\begin{array}{ll} 5 & 3 \\ -2 & -1 \end{array}\right]$

Hint : you must know the formula $\left(A^{T}\right)^{-1}$

Solution : We know that

$\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$

_{$\left(A^{T}\right)^{-1}=\left[\begin{array}{ll} 5 & 3 \\ -2 & -1 \end{array}\right]^{T}$}

_{$\left(A^{T}\right)^{-1}=\left[\begin{array}{ll} 5 & -2 \\ 3 & -1 \end{array}\right]$}

Adjoint and inverse of a Matrix Exercise very short answer type Question 8

Answer : $adj\left ( AB \right )= adjB.adjA$
Given :
Hint : use the formula $adj\left ( AB \right )= adjB.adjA$
Solution : $adj\left ( AB \right )= adjB.adjA$
$\begin{aligned} &=\left[\begin{array}{cc} 1 & -2 \\ -3 & 1 \end{array}\right]\left[\begin{array}{cc} 2 & 3 \\ 4 & -1 \end{array}\right] \\ &=\left[\begin{array}{cc} 2-8 & 3+2 \\ -6+4 & -9-1 \end{array}\right] \end{aligned}$
$\left ( adjAB \right )=\left[\begin{array}{cc} -6 & 5 \\ -2 & -10 \end{array}\right]$

Adjoint and inverse of a Matrix Exercise very short answer type Question 9

Answer : symmetric matrix
Given : A is a symmetric matrix
Hint : A is a symmetric matrix when
$A=A^{T}$
$A^{T}$ is symmetric when $A^{T}=\left ( A^{T} \right )^{T}$
So, we do transpose of $A^{T}$
$\left ( A^{T} \right )^{T}$
$=A^{TT}$
$=A$
$=A^{T}$ $\left ( A=A^{T} \right )$
Hence, $A^{T}$ is a symmetric matrix
$\text { [e.g. } \left.\left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right]^{\top}=\left[\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right]\right]$

Adjoint and inverse of a Matrix Exercise very short answer type Question 10

Answer : $2A$
Given : $n=3$ & $\left | A \right |=2$
Hint : use the formula of $adj\left ( adjA \right )$
Solution : We know that
$adj\left ( adjA \right )=\left | A \right |^{n-2}A$
$=\left ( 2 \right )^{3-2}A\left ( n=3,\left | a \right |=2 \right )$
$adj\left ( adjA \right )=2A$

Adjoint and inverse of a Matrix Exercise very short answer type Question 11

Answer : 81
Given : A is square matrix of order 3 such that $\left | A \right |=3$
Hint : use the formula of $adj\left ( adjA \right )$
Solution : $\left | A \right |=3$
$\left | adj\left ( adjA \right ) \right |=\left | A \right |^{\left ( n-1 \right )^{2}}$
$=3^{\left ( 3-1 \right )^{2}}$ (n=3) & $\left ( \left | A \right |=3 \right )$
$=3^{4}$
$\left |adj\left ( adjA \right ) \right |=81$

Adjoint and inverse of a Matrix Exercise very short answer type Question 12

Answer : 4
Given : adj(2A) = K adj(A) and A is a square matrix of order 3
Hint : you must know about the concept of square matrix
Solution : We know that
$\begin{aligned} &\operatorname{adj}(K A)=K^{n-1} \operatorname{adj}(A) \\ &\operatorname{adj}(2 A)=2^{3-1} \operatorname{adj}(A) \quad(n=3) \\ &\operatorname{adj}(2 A)=4 a d j A \\ &K=4 \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 13

Answer : null matrix
Given : A is a square matrix
Hint : use the formula of $adj\left ( A^{T} \right )$ & $adj\left ( A \right )^{T}$
Solution : We know that
$\begin{aligned} &\operatorname{adj}\left(A^{T}\right)=\left|A^{T}\right|\left(A^{T}\right)^{-1}\\ &\operatorname{adj}(A)^{T}=\left(|A| A^{-1}\right)^{T}\\ &\text { Now, we have }\\ &\operatorname{adj}\left(A^{T}\right)-\operatorname{adj}(A)^{T}\\ &=\left|A^{T}\right|\left(A^{T}\right)^{-1}-\left[|A| A^{-1}\right]^{T}\\ &=0 \quad \end{aligned}$ $\left[\because A^{\top} \cdot A^{-\top}=1\right]$

Adjoint and inverse of a Matrix Exercise very short answer type Question 14

Answer : 4
Given : n=3 and $A\left ( adjA \right )=2I$
Hint : use the formula of $\left | adjA \right |$
Solution : We know that
$\begin{aligned} &A(\operatorname{adj} A)=|A| I \\ &A(\operatorname{adj} A)=2 I \\ &|A|=2 \\ &\text { Now, }|\operatorname{adj} A|=|A|^{n-1} \\ &=(2)^{3-1} \\ &=(2)^{2} \\ &=4 \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 15

Answer : symmetric
Given : $A\neq 0$ & $A^{T}=A$
Hint : use the property of $\left ( A^{T} \right )^{-1}=\left ( A^{-1} \right )^{T}$
Solution : $\left ( A^{T} \right )^{-1}=\left ( A^{-1} \right )^{T}$
$\left ( A^{-1} \right )^{T}=A^{-1}\left ( A^{T}=A \right )$
$\left ( A^{-1} \right )^{T}=A^{-1}$
Hence,$A^{-1}$ is a symmetric

Adjoint and inverse of a Matrix Exercise very short answer type Question 16

Answer : 1
Given : $A=\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right] \& A(\operatorname{adj} A)=\left[\begin{array}{cc} K & 0 \\ 0 & K \end{array}\right]$
Hint : Find $A^{-1}$ and then multiply both side by A
Solution : $A\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$
$\begin{aligned} &|A|=\cos ^{2} \theta+\sin ^{2} \theta \\ &=1 \\ &|A| \neq 0 \\ &A^{-1} \text { exists } \\ &A^{-1}=\frac{a d j A}{|A|} \\ &A^{-1}=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right] \\ &A^{-1}=a d j A \end{aligned}$
multiply both side by A
$\begin{aligned} &A A^{-1}=A \operatorname{adj} A \\ &I=\left[\begin{array}{cc} K & 0 \\ 0 & K \end{array}\right] \\ &{\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]=\left[\begin{array}{cc} K & 0 \\ 0 & K \end{array}\right]} \\ &K=1(\text { \text{On comparing }}) \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 17

Answer : Given : A is an invertible matrix and $\left | A^{-1} \right |=2$
Hint : use the relation between $\left | A^{-1} \right |$ and $\left | A \right |$ of find the answer
Solution : We know that
$\begin{aligned} &\left|A^{-1}\right|=\frac{1}{|A|} \\ &2=\frac{1}{|A|}\left(\left|A^{-1}\right|=2\right) \\ &|A|=\frac{1}{2} \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 18

Answer : 25
Given: $A(\operatorname{adj} A)=\left[\begin{array}{lll} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{array}\right]$
Hint : Use the formula of A$A\left ( adjA \right )=\left | A \right |I$
Solution : $A(\operatorname{adj} A)=\left[\begin{array}{lll} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{array}\right]$
$\begin{aligned} &A(\operatorname{adj} A)=5 I \\ &|A|=5 \\ &N o w|\operatorname{adj} A|=|A|^{n-1} \\ &=(5)^{3-1}(n=3,|A|=5) \\ &=25 \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 19

Answer : $\frac{1}{19}$
Given : $A=\left[\begin{array}{cc} 2 & 3 \\ 5 & -2 \end{array}\right] \& A^{-1}=K A$
Hint : Find $A^{-1}$ and put it in equation and then compare it.
Solution : $$$ A=\left[\begin{array}{cc} 2 & 3 \\ 5 & -2 \end{array}\right]$
$$$ \begin{aligned} &|A|=-4-15 \\ &=-19 \\ &|A| \neq 0 \\ &A^{-1} \text { exists } \\ &A^{-1}=\frac{\operatorname{adj} A}{|A|} \end{aligned}$
$$$ \begin{aligned} &=\frac{-1}{19}\left[\begin{array}{cc} -2 & -3 \\ -5 & 2 \end{array}\right]\\ &=\frac{-1}{19}\left[\begin{array}{cc} 2 & 3 \\ 5 & -2 \end{array}\right]\\ &\text { Now } \mathrm{A}^{-1}=K A\\ &\frac{1}{19}\left[\begin{array}{cc} 2 & 3 \\ 5 & -2 \end{array}\right]=K\left[\begin{array}{cc} 2 & 3 \\ 5 & -2 \end{array}\right]\\ &\text { On comparing }\\ &\mathrm{K}=\frac{1}{19} \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 20

Answer : $A^{-1}=I-A$
Given : $A^{2}-A+I=0$
Hint : Use pre-multiplication technique
Solution : $A^{2}-A+I=0$
$A^{2}-A=-I$
pre-multiplication by $A^{-1}$ on both sides
$$$ \begin{aligned} &A\left(A A^{-1}\right)-A A^{-1}=-I A^{-1} \\ &A-I=-A^{-1} \\ &A^{-1}=I-A \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 21

Answer : 110
Given : $$$ A=\left[\begin{array}{ccc} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array}\right]$
Hint : you must know about how to calculate cofactor of an element
Solution : $$$ A=\left[\begin{array}{ccc} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array}\right]$
$$$ \begin{aligned} &c_{32}=(-1)^{3+2}(8-30) \\ &=-1(-22) \\ &=22 \\ &a_{32}=5 \\ &a_{32} c_{32}=22(5) \\ &=110 \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 22

Answer : $\left[\begin{array}{cc} 5 & 2 \\ 7 & 3 \end{array}\right]$
Given : $\left[\begin{array}{cc} 3 & -2 \\ -7 & 5 \end{array}\right]$
Hint : you must know about how to find inverse of a matrix
Solution : $\left[\begin{array}{cc} 3 & -2 \\ -7 & 5 \end{array}\right]$
$$$ \begin{aligned} &c_{11}=5, c_{12}=7, c_{21}=2, c_{22}=3 \\ &=\frac{1}{1}\left[\begin{array}{cc} 5 & 7 \\ 2 & 3 \end{array}\right] \\ &=\left[\begin{array}{cc} 5 & 7 \\ 2 & 3 \end{array}\right] \end{aligned}$
Now transpose it, we get

Adjoint and inverse of a Matrix Exercise very short answer type Question 23

Answer : $$$ \left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$
Given : $$$ \left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$
$$$ \left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$
Hint : Find the cofactor and then transpose it
Solution : $$$ \left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$
$$$ \left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$
$$$ \begin{aligned} &c_{11}=\cos \theta, c_{12}=\sin \theta, c_{21}=-\sin \theta, c_{22}=\cos \theta \\ &|A|=\cos ^{2} \theta-\left(-\sin ^{2} \theta\right) \\ &=\cos ^{2} \theta+\sin ^{2} \theta \quad \end{aligned}$ $\left[\because \cos ^{2} \theta+\sin ^{2} \theta=1\right] \\$
$=1$
$A^{-1}=\frac{A d j|A|}{|A|}=\frac{1}{1}$$\left[\begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$
Now transpose it, we get
$=\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$

Adjoint and inverse of a Matrix Exercise very short answer type Question 24

Answer : $\left[\begin{array}{cc} 0 & 3 \\ -2 & 1 \end{array}\right]$
Given : $\left[\begin{array}{cc} 1 & -3 \\ 2 & 0 \end{array}\right]$
Hint : find the cofactors to find adj A
solution :
$A=\left[\begin{array}{cc} 0 & -2 \\ 3 & 1 \end{array}\right]$
$=\left[\begin{array}{cc} 0 & 3 \\ -2 & 1 \end{array}\right]$ $[ \text{co-factors of element A are, }\left.C_{11}=0, C_{12}=-2, C_{21}=3, C_{22}=1\right]$

Adjoint and inverse of a Matrix Exercise very short answer type Question 25

Answer : $\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]$
Given : $A=\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]$ &B=$\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$
Hint : find adj A and find adj B and then use the formula of adj (AB)
Solution : $A=\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]$
$adj A=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]$
$B=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$
$adj B=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$
Now we know that
adj(AB) =adj(B)adj(A)
$=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]$
$adj\left ( AB \right )=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]$

Adjoint and inverse of a Matrix Exercise very short answer type Question 26

Anawer : -11
Given : $A=\left[\begin{array}{cc}3 & 1 \\ 2 & -3\end{array}\right]$
Hint : find $\left | A \right |$ and then use the formula of $\left | adjA \right |$
Solution : $A=\left[\begin{array}{cc}3 & 1 \\ 2 & -3\end{array}\right]$
$\left | A \right |=-9-2$
=-11
Now ,$\left | adjA \right |=\left | A \right |^{n-1}$
$=\left ( -11 \right )^{2-1}$
$=-11$

Adjoint and inverse of a Matrix Exercise very short answer type Question 27

Answer : $A^{-1}=\frac{1}{19}A$
Given : $A.=\left[\begin{array}{cc} 2 & 3 \\ 5 & -2 \end{array}\right]$
Hint : to find $A^{-1^{}}$ ,find adj A & $\left | A \right |$
Solution : $A.=\left[\begin{array}{cc} 2 & 3 \\ 5 & -2 \end{array}\right]$
$\left | A \right |=-4-15$
$=-19$
$\left | A \right |\neq 0$
So, $A^{-1}$exists
$adjA=\left[\begin{array}{cc} -2 & -5 \\ -3 & 2 \end{array}\right]^{T}$
$=\left[\begin{array}{cc} -2 & -3 \\ -5 & -2 \end{array}\right]$
Now $A^{-1}=\frac{adjA}{\left | A \right |}$
$A^{-1}=-\frac{1}{19}$ $\left[\begin{array}{cc} -2 & -3 \\ -5 & -2 \end{array}\right]$
$A^{-1}=\frac{1}{19}\left[\begin{array}{cc} 2 & 3 \\ 5 & -2 \end{array}\right]$
$A^{-1}=\frac{1}{19}A$

Adjoint and inverse of a Matrix Exercise very short answer type Question 28

Answer : $A^{-1}=\left[\begin{array}{cc} 3 & -5 \\ -1 & 2 \end{array}\right]$
Given : $A=\left[\begin{array}{cc} 2 & 5 \\ 1 & 3 \end{array}\right]$
Hint : find adj A and $\left | A \right |$ and then put in $A^{-1}$ formula
Solution : $A=\left[\begin{array}{cc} 2 & 5 \\ 1 & 3 \end{array}\right]$
$\left | A \right |=6-5$
$=1$
$\left | A \right |$$\neq 0$
So ,$A^{-1}$ exists
$adjA=\left[\begin{array}{cc} 3 & -1 \\ -5 & 2 \end{array}\right]^{T}$
$=\left[\begin{array}{cc} 3 & -5 \\ -1 & 2 \end{array}\right]$
$A^{-1}=\frac{adjA}{\left | A \right |}$
$A^{-1}=\frac{1}{1}\left[\begin{array}{cc} 3 & -5 \\ -1 & 2 \end{array}\right]$
$A^{-1}=\left[\begin{array}{cc} 3 & -5 \\ -1 & 2 \end{array}\right]$

Adjoint and inverse of a Matrix Exercise very short answer type Question 29

Answer : $\left[\begin{array}{ll} 2 & 5 \\ 5 & 4 \end{array}\right]=\left[\begin{array}{ll} 3 & 1 \\ 2 & 0 \end{array}\right]\left[\begin{array}{cc} 1 & 2 \\ -1 & -1 \end{array}\right]$
Given : $\left[\begin{array}{ll} 2 & 1 \\ 2 & 0 \end{array}\right]=\left[\begin{array}{ll} 3 & 1 \\ 2 & 0 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ -1 & 1 \end{array}\right]$
and column operation $C_{2} \rightarrow C_{2}+2 C_{1} \\$
Hint : you must know about the concept of elementary transform
Solution : The Matrix is in the form $X=AB$
Elementary opration can be apply only on X and B
$\begin{aligned} &\text { So },\left[\begin{array}{ll} 2 & 1 \\ 2 & 0 \end{array}\right]=\left[\begin{array}{ll} 3 & 1 \\ 2 & 0 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ -1 & 1 \end{array}\right] \\ &C_{2} \rightarrow C_{2}+2 C_{1} \\ &{\left[\begin{array}{ll} 2 & 1+4 \\ 2 & 0+4 \end{array}\right]=\left[\begin{array}{ll} 3 & 1 \\ 2 & 0 \end{array}\right]\left[\begin{array}{cc} 1 & 0+2 \\ -1 & 1-2 \end{array}\right]} \\ &{\left[\begin{array}{ll} 2 & 5 \\ 2 & 4 \end{array}\right]=\left[\begin{array}{cc} 3 & 1 \\ 2 & 0 \end{array}\right]\left[\begin{array}{cc} 1 & 2 \\ -1 & -1 \end{array}\right]} \end{aligned}$

Adjoint and inverse of a Matrix Exercise very short answer type Question 30

Answer : $\left[\begin{array}{ll} 2 & 3 \\ 3 & 7 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 2 & -1 \end{array}\right]=\left[\begin{array}{cc} 8 & -3 \\ 17 & -7 \end{array}\right]$
Given : $\left[\begin{array}{ll} 2 & 3 \\ 1 & 4 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 2 & -1 \end{array}\right]=\left[\begin{array}{cc} 8 & -3 \\ 9 & -4 \end{array}\right]$
and elementary operation $R_{2}\rightarrow R_{2}+R_{1}$
Hint : In multiplication row operation is equivalent to left multiplication
Solution :
$\left[\begin{array}{ll} 2 & 3 \\ 1 & 4 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 2 & -1 \end{array}\right]=\left[\begin{array}{cc} 8 & -3 \\ 9 & -4 \end{array}\right]$

By applying operation $R_{2}\rightarrow R_{2}+R_{1}$
$\left[\begin{array}{ll} 2 & 3 \\ 3 & 7 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 2 & -1 \end{array}\right]=\left[\begin{array}{cc} 8 & -3 \\ 17 & -7 \end{array}\right]$

Adjoint and inverse of a Matrix Exercise very short answer type Question 31

Answer : 4
Given $\left | A \right |=4$
Hint : use the concept of A adj A
Solution : $\left | A adj A\right |=?$
$\begin{aligned} &A(\operatorname{adj} A)=|A| I \\ &A(\operatorname{adj} A)=4 I \quad[|A|=4] \\ &|A(\operatorname{adj} A)|=|4 I| \\ &|A(\operatorname{adj} A)|=4\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \\ &=4 \times 1 \\ &=4 \\ &|A(\operatorname{adj} A)|=4 \end{aligned}$

RD Sharma Class 12th Chapter 6 VSA contains a total of 31 short answer questions from the chapter 'Adjoint and Inverse of a Matrix' that are basic and easy to understand. It covers simple concepts like the square of a matrix, multiplication of matrices, transpose, the inverse of matrices, and other algebraic problems. Once students are through with this chapter's initial concepts, they can breeze through these solutions with ease.

RD Sharma Class 12th Chapter 6 VSA provided by Brainly contains solved questions from the RD Sharma book. This will help students save a lot of time preparing and enable them to understand the subject better. ‘Matrices’ is a vast topic that contains hundreds of questions.

As it is not possible for teachers to cover the entirety of the chapter through lectures, Brainly has provided this material for students' reference. The advantages of using RD Sharma Class 12th Chapter 6 VSA material are:

1. Expert created solutions

The solutions provided in RD Sharma Class 12th Chapter 6 VSA material are accurate, easy to understand, and on par with the CBSE syllabus as experts create them. Moreover, these solutions are exam-oriented and will help the students write better answers.

2. Easy for revision purposes

If students face a problem with any question from the RD Sharma book, they can refer to RD Sharma Class 12 Chapter 6 VSA material for revision as it contains solved questions.

3. Learn effectively from anywhere

Students can download the material from Brainly's website as a PDF which means they can access it offline from anywhere and any device. Physical hard copies are difficult to carry and maintain. Instead, students can download this material on their device for easy preparation. .

4. Stay ahead of the class

Students can use Class 12 RD Sharma chapter 6 VSA Solution as a tool to track their progress in the subject. They can even stay ahead of their class lectures by preparing well ahead of time. This will help them understand the concepts better and save a lot of time for preparation and revision.

5. Free to download

RD Sharma Class 12th Chapter 6 VSA is available for all students for free on Brainly's website. Students have to search the book name and exercise on the website, and it will lead them straight to the material.