ML Aggarwal Solution Class 10 Chapter 8 Matrices Exercise 8.2

 Exercise 8.2

Question 1

Given that $M=\left[\begin{array}{ll}2 & 0 \\ 1 & 2\end{array}\right]$ and $N=\left[\begin{array}{cc}2 & 0 \\ -1 & 2\end{array}\right]$,find M+2N

Sol :

$M=\left[\begin{array}{ll}2 & 0 \\ 1 & 2\end{array}\right]$

$N=\left[\begin{array}{cc}2 & 0 \\ -1 & 2\end{array}\right]$

$\therefore \mathrm{M}+2 \mathrm{~N}=\left[\begin{array}{ll}2 & 0 \\ 1 & 2\end{array}\right]+2\left[\begin{array}{rr}2 & 0 \\ -1 & 2\end{array}\right]$

$=\left[\begin{array}{ll}2 & 0 \\ 1 & 2\end{array}\right]+\left[\begin{array}{rr}4 & 0 \\ -2 & 4\end{array}\right]$

$=\left[\begin{array}{ll}2+4 & 0+0 \\ 1-2 & 2+4\end{array}\right]=\left[\begin{array}{rr}6 & 0 \\ -1 & 6\end{array}\right]$

Question 2

If $A=\left[\begin{array}{cc}2 & 0 \\ -3 & 1\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & 1 \\ -2 & 3\end{array}\right]$

find 2A – 3B

Sol :

$A=\left[\begin{array}{cc}2 & 0 \\ -3 & 1\end{array}\right]$

$B=\left[\begin{array}{cc}0 & 1 \\ -2 & 3\end{array}\right]$

$\therefore 2 \mathrm{~A}-3 \mathrm{~B}=2\left[\begin{array}{rr}2 & 0 \\ -3 & 1\end{array}\right]-3\left[\begin{array}{rr}0 & 1 \\ -2 & 3\end{array}\right]$

$=\left[\begin{array}{rr}4 & 0 \\ -6 & 2\end{array}\right]-\left[\begin{array}{rr}0 & 3 \\ -6 & 9\end{array}\right]=\left[\begin{array}{rr}4-0 & 0-3 \\ -6+6 & 2-9\end{array}\right]$

$=\left[\begin{array}{ll}4 & -3 \\ 0 & -7\end{array}\right]$

Question 3

If $A=\left[\begin{array}{ll}1 & 4 \\ 2 & 3\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 2 \\ 3 & 1\end{array}\right]$

Compute 3A + 4B

Sol :

$A=\left[\begin{array}{ll}1 & 4 \\ 2 & 3\end{array}\right]$

$B=\left[\begin{array}{ll}1 & 2 \\ 3 & 1\end{array}\right]$

$3 \mathrm{~A}+4 \mathrm{~B}=3\left[\begin{array}{ll}1 & 4 \\ 2 & 3\end{array}\right]+4\left[\begin{array}{ll}1 & 2 \\ 3 & 1\end{array}\right]$

$=\left[\begin{array}{rr}3 & 12 \\ 6 & 9\end{array}\right]+\left[\begin{array}{rr}4 & 8 \\ 12 & 4\end{array}\right]$

$=\left[\begin{array}{rr}3+4 & 12+8 \\ 6+12 & 9+4\end{array}\right]=\left[\begin{array}{rr}7 & 20 \\ 18 & 13\end{array}\right]$

Question 4

Given $A=\left[\begin{array}{ll}1 & 4 \\ 2 & 3\end{array}\right]$ and $B=\left[\begin{array}{cc}-4 & -1 \\ -3 & -2\end{array}\right]$

(i) find the matrix 2A + B

(ii) find a matrix C such that C + B = $\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

Sol :

$A=\left[\begin{array}{ll}1 & 4 \\ 2 & 3\end{array}\right]$

$B=\left[\begin{array}{ll}-4 & -1 \\ -3 & -2\end{array}\right]$

(i) $2 \mathrm{~A}+\mathrm{B}=2\left[\begin{array}{ll}1 & 4 \\ 2 & 3\end{array}\right]+\left[\begin{array}{ll}-4 & -1 \\ -3 & -2\end{array}\right]$

$=\left[\begin{array}{ll}2 & 8 \\ 4 & 6\end{array}\right]+\left[\begin{array}{ll}-4 & -1 \\ -3 & -2\end{array}\right]$

$=\left[\begin{array}{ll}2-4 & 8-1 \\ 4-3 & 6-2\end{array}\right]=\left[\begin{array}{rr}-2 & 7 \\ 1 & 4\end{array}\right]$

(ii) $\mathrm{C}+\mathrm{B}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

$\mathrm{C}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]-\mathrm{B}=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]-\left[\begin{array}{ll}-4 & -1 \\ -3 & -2\end{array}\right]$

$=\left[\begin{array}{ll}0-(-4) & 0-(-1) \\ 0-(-3) & 0-(-2)\end{array}\right]=\left[\begin{array}{ll}4 & 1 \\ 3 & 2\end{array}\right]$

Question 5

$A=\left[\begin{array}{cc}1 & 2 \\ -2 & 3\end{array}\right]$ and $B=\left[\begin{array}{cc}-2 & -1 \\ 1 & 2\end{array}\right], C=\left[\begin{array}{cc}0 & 3 \\ 2 & -1\end{array}\right]$

Find A + 2B – 3C

Sol :

$A=\left[\begin{array}{cc}1 & 2 \\ -2 & 3\end{array}\right]$ and $B=\left[\begin{array}{cc}-2 & -1 \\ 1 & 2\end{array}\right], C=\left[\begin{array}{cc}0 & 3 \\ 2 & -1\end{array}\right]$

∴ A + 2B – 3C

$=\left[\begin{array}{rr}1 & 2 \\ -2 & 3\end{array}\right]+2\left[\begin{array}{rr}-2 & -1 \\ 1 & 2\end{array}\right]-3\left[\begin{array}{rr}0 & 3 \\ 2 & -1\end{array}\right]$

$=\left[\begin{array}{rr}1 & 2 \\ -2 & 3\end{array}\right]+\left[\begin{array}{rr}-4 & -2 \\ 2 & 4\end{array}\right]-\left[\begin{array}{rr}0 & 9 \\ 6 & -3\end{array}\right]$

$=\left[\begin{array}{rr}1-4-0 & 2-2-9 \\ -2+2-6 & 3+4+3\end{array}\right]=\left[\begin{array}{rr}-3 & -9 \\ -6 & 10\end{array}\right]$

Question 6

If $A=\left[\begin{array}{cc}0 & -1 \\ 1 & 2\end{array}\right]$ and $B=\left[\begin{array}{cc}1 & 2 \\ -1 & 1\end{array}\right]$

Find the matrix X if :

(i) 3A + X = B

(ii) X – 3B = 2A

Sol :

$A=\left[\begin{array}{ll}0 & -1 \\ 1 & 2\end{array}\right]$

$B=\left[\begin{array}{cc}1 & 2 \\ -1 & 1\end{array}\right]$

(i) 3A + X = B

⇒ X = B – 3A

$X=\left[\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right]-3\left[\begin{array}{rr}0 & -1 \\ 1 & 2\end{array}\right]$

$=\left[\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right]-\left[\begin{array}{rr}0 & -3 \\ 3 & 6\end{array}\right]$

$=\left[\begin{array}{rr}1-0 & 2+3 \\ -1-3 & 1-6\end{array}\right]=\left[\begin{array}{rr}1 & 5 \\ -4 & -5\end{array}\right]$

Question 7

Solve the matrix equation

$\left[\begin{array}{ll}2 & 1 \\ 5 & 0\end{array}\right]-3 X=\left[\begin{array}{cc}-7 & 4 \\ 2 & 6\end{array}\right]$

Sol :

$\left[\begin{array}{ll}2 & 1 \\ 5 & 0\end{array}\right]-3 X=\left[\begin{array}{cc}-7 & 4 \\ 2 & 6\end{array}\right]$

$\left[\begin{array}{ll}2 & 1 \\ 5 & 0\end{array}\right]-\left[\begin{array}{cc}-7 & 4 \\ 2 & 6\end{array}\right]=3 X$

$\therefore X=\frac{1}{3}\left[\begin{array}{ll}9 & -3 \\ 3 & -6\end{array}\right]=\left[\begin{array}{ll}3 & -1 \\ 1 & -2\end{array}\right]$

Question 8

If $\left[\begin{array}{cc}1 & 4 \\ -2 & 3\end{array}\right]+2 M=3\left[\begin{array}{rr}3 & 2 \\ 0 & -3\end{array}\right]$, find the matrix M

Sol :

$\left[\begin{array}{cc}1 & 4 \\ -2 & 3\end{array}\right]+2 M=3\left[\begin{array}{cc}3 & 2 \\ 0 & -3\end{array}\right]$

2M =

$3\left[\begin{array}{cc}3 & 2 \\ 0 & -3\end{array}\right]-\left[\begin{array}{cc}1 & 4 \\ -2 & 3\end{array}\right]=\left[\begin{array}{cc}9 & 6 \\ 0 & -9\end{array}\right]-\left[\begin{array}{cc}1 & 4 \\ -2 & 3\end{array}\right]$

$=\left[\begin{array}{cc}9-1 & 6-4 \\ 0-(-2) & -9-3\end{array}\right]=\left[\begin{array}{cc}8 & 2 \\ 2 & -12\end{array}\right]$

$\therefore M=\frac{1}{2}\left[\begin{array}{cc}8 & 2 \\ 2 & -12\end{array}\right]=\left[\begin{array}{cc}4 & 1 \\ 1 & -6\end{array}\right]$

(Dividing by 2)

Question 9

$A=\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]$ and $B=\left[\begin{array}{cc}-3 & 2 \\ 4 & 0\end{array}\right], C=\left[\begin{array}{ll}4 & 0 \\ 0 & 2\end{array}\right]$

Find the matrix X such that A + 2X = 2B + C

Sol :

$A=\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]$ and $B=\left[\begin{array}{cc}-3 & 2 \\ 4 & 0\end{array}\right], C=\left[\begin{array}{cc}4 & 0 \\ 0 & 2\end{array}\right]$

Iet $X=\left[\begin{array}{ll}x & y \\ z & t\end{array}\right]$

A+2 X=2 B+C

2 X=2 B+C-A

$2\left[\begin{array}{ll}x & y \\ z & t\end{array}\right]=2\left[\begin{array}{cc}-3 & 2 \\ 4 & 0\end{array}\right]+\left[\begin{array}{ll}4 & 0 \\ 0 & 2\end{array}\right]-\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]$

$=\left[\begin{array}{cc}-6 & 4 \\ 8 & 0\end{array}\right]+\left[\begin{array}{cc}4 & 0 \\ 0 & 2\end{array}\right]-\left[\begin{array}{cc}2 & -6 \\ 2 & 0\end{array}\right]$

$=\left[\begin{array}{cc}-6+4-2 & 4+0+6 \\ 8+0-2 & 0+2-0\end{array}\right]=\left[\begin{array}{cc}-4 & -10 \\ 6 & 2\end{array}\right]$

$\therefore 2\left[\begin{array}{ll}x & y \\ z & t\end{array}\right]=\left[\begin{array}{cc}-4 & 10 \\ 6 & 2\end{array}\right]$

$\therefore\left[\begin{array}{ll}x & y \\ z & t\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}-4 & 10 \\ 6 & 2\end{array}\right]=\left[\begin{array}{cc}-2 & 5 \\ 3 & 1\end{array}\right]$

Question 10

Find X and Y if X + Y = $\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]$ and $X-Y=\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$

Sol :

$X+Y=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]$...(i)

$X-Y=\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$...(ii)

Adding (i) and (ii) we get, $2 x=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]+\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$

$=\left[\begin{array}{ll}7+3 & 0+0 \\ 2+0 & 5+3\end{array}\right]=\left[\begin{array}{rr}10 & 0 \\ 2 & 8\end{array}\right]$

$\therefore x=\frac{1}{2}\left[\begin{array}{rr}10 & 0 \\ 2 & 8\end{array}\right]=\left[\begin{array}{ll}5 & 0 \\ 1 & 4\end{array}\right]$

Subtracting (ii) from (i) 

$2 y=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$

$\Rightarrow 2 y=\left[\begin{array}{ll}7-3 & 0-0 \\ 2-0 & 5-3\end{array}\right]=\left[\begin{array}{ll}4 & 0 \\ 2 & 2\end{array}\right]$

$\therefore y=\frac{1}{2}\left[\begin{array}{ll}4 & 0 \\ 2 & 2\end{array}\right]=\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]$

Hence, $x=\left[\begin{array}{ll}5 & 0 \\ 1 & 4\end{array}\right], y=\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right]$

Question 11

If $2\left[\begin{array}{ll}3 & 4 \\ 5 & x\end{array}\right]+\left[\begin{array}{ll}1 & y \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}7 & 0 \\ 10 & 5\end{array}\right]$  Find the values of x and y

Sol :

$2\left[\begin{array}{ll}3 & 4 \\ 5 & x\end{array}\right]+\left[\begin{array}{ll}1 & y \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}7 & 0 \\ 10 & 5\end{array}\right]$

$\left[\begin{array}{cc}6 & 8 \\ 10 & 2 x\end{array}\right]+\left[\begin{array}{ll}1 & y \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}7 & 0 \\ 10 & 5\end{array}\right]$

$\Rightarrow\left[\begin{array}{rr}6+1 & 8+y \\ 10+0 & 2 x+1\end{array}\right]=\left[\begin{array}{rr}7 & 0 \\ 10 & 5\end{array}\right]$

Comparing the corresponding elements,

8+y=0 then y=-8

2x+1=5 then 2x=5-1=4 

$\Rightarrow x=2$

Hence, x=2, y=-8

Question 12

If $2\left[\begin{array}{ll}3 & 4 \\ 5 & x\end{array}\right]+\left[\begin{array}{ll}1 & y \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}z & 0 \\ 10 & 5\end{array}\right]$ Find the values of x and y

Sol :

$2\left[\begin{array}{ll}3 & 4 \\ 5 & x\end{array}\right]+\left[\begin{array}{ll}1 & y \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}z & 0 \\ 10 & 5\end{array}\right]$

$\left[\begin{array}{cc}6 & 8 \\ 10 & 2 x\end{array}\right]+\left[\begin{array}{ll}1 & y \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}z & 0 \\ 10 & 5\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}6+1 & 8+y \\ 10+0 & 2 x+1\end{array}\right]=\left[\begin{array}{cc}z & 0 \\ 10 & 5\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}7 & 8+y \\ 10 & 2 x+1\end{array}\right]=\left[\begin{array}{cc}z & 0 \\ 10 & 5\end{array}\right]$

Comparing, 

2x+1=5 

$\Rightarrow 2 x=5-1=4$

$\therefore x=\frac{4}{2}=2$

8+y=0,

$\Rightarrow y=-8$

z=7

Hence x=2, y=-8, z=7

Question 13

If $\left[\begin{array}{cc}5 & 2 \\ -1 & y+1\end{array}\right]-2\left[\begin{array}{cc}1 & 2 x-1 \\ 3 & -2\end{array}\right]=\left[\begin{array}{cc}3 & -8 \\ -7 & 2\end{array}\right]$ Find the values of x and y

Sol :

$\left[\begin{array}{cc}5 & 2 \\ -1 & y+1\end{array}\right]-2\left[\begin{array}{cc}1 & 2 x-1 \\ 3 & -2\end{array}\right]=\left[\begin{array}{cc}3 & -8 \\ -7 & 2\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}5 & 2 \\ -1 & y+1\end{array}\right]-\left[\begin{array}{cc}2 & 4 x-2 \\ 6 & -4\end{array}\right]=\left[\begin{array}{cc}3 & -8 \\ -7 & 2\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}5-2 & 2-4 x+2 \\ -1-6 & y+1+4\end{array}\right]=\left[\begin{array}{cc}3 & -8 \\ -7 & 2\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}3 & 4-4 x \\ -7 & y+5\end{array}\right]=\left[\begin{array}{cc}3 & -8 \\ -7 & 2\end{array}\right]$

Comparing the corresponding terms, we get 

4-4 x=-8 

$\Rightarrow-4 x=-8-4$

$\Rightarrow-4 x=-12$

$\Rightarrow x=\frac{-12}{-4}=3$

and y+5=2

$\Rightarrow y=2-5=-3$

$\therefore x=3, y=-3$

Question 14

If $\left[\begin{array}{ll}a & 3 \\ 4 & 2\end{array}\right]+\left[\begin{array}{rr}2 & b \\ 1 & -2\end{array}\right]-\left[\begin{array}{cc}1 & 1 \\ -2 & c\end{array}\right]=\left[\begin{array}{ll}5 & 0 \\ 7 & 3\end{array}\right]$

Find the value of a,b and c

Sol :

$\left[\begin{array}{ll}a & 3 \\ 4 & 2\end{array}\right]+\left[\begin{array}{rr}2 & b \\ 1 & -2\end{array}\right]-\left[\begin{array}{cc}1 & 1 \\ -2 & c\end{array}\right]=\left[\begin{array}{ll}5 & 0 \\ 7 & 3\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}a+2-1 & 3+b-1 \\ 4+1+2 & 2-2-c\end{array}\right]=\left[\begin{array}{ll}5 & 0 \\ 7 & 3\end{array}\right]$

$\Rightarrow\left[\begin{array}{rr}a+1 & b+2 \\ 7 & -c\end{array}\right]=\left[\begin{array}{ll}5 & 0 \\ 7 & 3\end{array}\right]$

Comparing the corresponding elements

$\begin{aligned} a+1 &=5 \Rightarrow a=4 \\ b+2 &=0 \Rightarrow b=-2 \\-c=3 & \Rightarrow c=-3  \end{aligned}$

Question 15

If $A=\left[\begin{array}{cc}2 & a \\ -3 & 5\end{array}\right]$ and $B=\left[\begin{array}{cc}-2 & 3 \\ 7 & b\end{array}\right], C=\left[\begin{array}{cc}c & 9 \\ -1 & -11\end{array}\right]$ and 5A+2B=C, find the value of a,b,c

Sol :

$A=\left[\begin{array}{cc}2 & a \\ -3 & 5\end{array}\right]$ and $B=\left[\begin{array}{cc}-2 & 3 \\ 7 & b\end{array}\right], C=\left[\begin{array}{cc}c & 9 \\ -1 & -11\end{array}\right]$

and 5A + 2B = C

$\Rightarrow 5\left[\begin{array}{cc}2 & a \\ -3 & 5\end{array}\right]+2\left[\begin{array}{cc}-2 & 3 \\ 7 & b\end{array}\right]=\left[\begin{array}{cc}c & 9 \\ -1 & -11\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}10 & 5 a \\ -15 & 25\end{array}\right]+\left[\begin{array}{cc}-4 & 6 \\ 14 & 2 b\end{array}\right]=\left[\begin{array}{cc}c & 9 \\ -1 & -11\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}10-4 & 5 a+6 \\ -15+14 & 25+2 b\end{array}\right]=\left[\begin{array}{cc}c & 9 \\ -1 & -11\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}6 & 5 a+6 \\ -1 & 25+2 b\end{array}\right]=\left[\begin{array}{cc}c & 9 \\ -1 & -11\end{array}\right]$

Comparing each term

5a+6=9

$\Rightarrow 5 a=9-6=3$

$\Rightarrow a=\frac{3}{5}$

$\Rightarrow 25+2 b=-11$

$\Rightarrow 2 b=-11-25=-36$

$\Rightarrow b=-\frac{36}{2}=-18$

c=6

Hence $a=\frac{3}{5}, b=-18$ and $c=6$