ML Aggarwal Solution Class 10 Chapter 8 Matrices Exercise 8.1

 Exercise 8.1

Question 1

(i) $\left[\begin{array}{cc}2 & -1 \\ 5 & 1\end{array}\right]$

(ii)[2 3 – 7]

(iii) $\left[\begin{array}{c}3 \\ 0 \\ -1\end{array}\right]$

(iv) $\left[\begin{array}{cc}2 & -4 \\ 0 & 0 \\ 1 & 7\end{array}\right]$

(v) $\left[\begin{array}{rll}2 & 7 & 8 \\ -1 & \sqrt{2} & 0\end{array}\right]$

(vi) $\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$

Sol :

(i) It is square matrix of order 2

(ii) It is row matrix of order 1 × 3

(iii) It is column matrix of order 3 × 1

(iv) It is matrix of order 3 × 2

(v) It is matrix of order 2 × 3

(vi) It is zero matrix of order 2 × 3

Question 2

(i) If a matrix has 4 elements, what are the possible order it can have ?

(ii) If a matrix has 8 elements, what are the possible order it can have ?

Sol :

(i) It can have 1 × 4, 4 × 1 or 2 × 2 order

(ii) It can have 1 × 8, 8 × 1,2 × 4 or 4 × 2 order

Question 3

Construct a 2 x 2 matrix whose elements aij are given by

(i) $a_{ij}=2 \mathrm{i}-j$

(ii) $\mathrm{a}_{ij}=\mathrm{i} . j$

Sol :

(i) It can be $\left[\begin{array}{ll}1 & 0 \\ 3 & 2\end{array}\right]$

(ii) It can be $\left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right]$

Question 4

Find the values of x and y if : $\left[\begin{array}{c}2 x+y \\ 3 x-2 y\end{array}\right]=\left[\begin{array}{l}5 \\ 4\end{array}\right]$

Sol :

Comparing corresponding elements,

2x + y = 5 …(i)

3x – 2y = 4 …(ii)

Multiply (i) by 2 and (ii) by ‘1’ we get

4x + 2y = 10, 3x – 2y = 4

Adding we get, 7x = 14 ⇒ x = 2

Substituting the value of x in (i)

2 x 2 + y = 5 ⇒ 4 + y = 5

y = 5 – 4 = 1

Hence x = 2, y = 1

Question 5

Find the value of $x$ if $\left[\begin{array}{cr}3 x+y & -y \\ 2 y-x & 3\end{array}\right]=\left[\begin{array}{cc}1 & 2 \\ -5 & 3\end{array}\right]$

Sol :

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

Comparing the corresponding terms, we get.

-y = 2

⇒ y = -2

$3 x+y=1 \Rightarrow 3 x+1-y$

$\Rightarrow 3 x=1-(-2)=1+2=3$

$\Rightarrow x=\frac{3}{3}=1$

Hence, x=1, y=-2

Question 6

If $\left[\begin{array}{cc}x+3 & 4 \\ y-4 & x+y\end{array}\right]=\left[\begin{array}{cc}5 & 4 \\ 3 & 9\end{array}\right]$, find values of x and y

Sol :

$\left[\begin{array}{cc}x+3 & 4 \\ y-4 & x+y\end{array}\right]=\left[\begin{array}{ll}5 & 4 \\ 3 & 9\end{array}\right]$

Comparing the corresponding terms, we get.

x + 3 = 5

⇒ x = 5 – 3 = 2

⇒ y – 4 = 3

⇒ y = 3 + 4 = 7

x = 2, y = 7

Question 7

Find the values of x, y and z if

$\left[\begin{array}{lr}x+2 & 6 \\ 3 & 5 z\end{array}\right]=\left[\begin{array}{cc}-5 & y^{2}+y \\ 3 & -20\end{array}\right]$

Sol :

Comparing the corresponding elements of equal determinents,

x + 2 = -5

⇒ x = -5 – 2 = -7

$\therefore x=-7,5 z=-20$

$\Rightarrow  z=-\frac{20}{5}=-4$

$\Rightarrow y^{2}+y=6$

$\Rightarrow y^{2}+y-6=0$

$\Rightarrow y^{2}+3 y-2 y-6=0$

$\Rightarrow y(y+3)-2(y+3)=0$

$\Rightarrow (y+3)(y-2)=0$

Either y+3=0

then y=-3 or y-2=0, then y=2

Hence, x=-7, y=-3,2,z=-4

Question 8

Find the values of x, y, a and b if

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

Sol :

Comparing corresponding elements

x – 2 = 3, y = 1

x = 3 + 2 = 5

a + 2b = 5 ……(i)

3a – b = 1 ……..(ii)

Multiplying (i) by 1 and (ii) by 2 

a+2 b=5, 6a-2b=2

Adding, we get 7a=7 

⇒ a=1

Substituting the value of a in..(i)

1+2 b=5 

⇒ 2b-5-1=4 

⇒ b=2

Hence x=5, y=1, a=1, b=2

Question 9

Find the values of a, b, c and d if

$\left[\begin{array}{cc}a+b & 3 \\ 5+c & a b\end{array}\right]=\left[\begin{array}{cc}6 & d \\ -1 & 8\end{array}\right]$

Sol :

$\left[\begin{array}{cc}a+b & 3 \\ 5+c & a b\end{array}\right]=\left[\begin{array}{cc}6 & d \\ -1 & 8\end{array}\right]$

Comparing the corresponding terms, we get.

3 = d ⇒ d = 3

⇒ 5 + c = – 1

⇒ c = -1 – 5

⇒ c = -6

a + b = 6 and ab = 8

$\therefore (a-b)^{2}=(a+b)^{2}-4 a b$

$=(6)^{2}-4 \times 8=36-32=4=(\pm 2)^{2}$

$\therefore a-b=\pm 2$

(i) If a-b=2

a+b=6

Adding , we get 2a=8

⇒a=4

a+b=6

⇒4+b=6

⇒b=6-4=2

∴a=4, b=2

(ii) If a-b=-2

a+b=6

Adding , we get 2a=4

$\Rightarrow a=\frac{4}{2}=2$

$a+b=6 \Rightarrow 2+b=6 \Rightarrow b=6^{2}-2=4$

$\therefore a=2, b=4$

Hence $a=4, b=2,$ or $a=2, b=4$

$c=-6$ and $d=3$

Question 10

Find the values of x, y, a and b, if

$\left[\begin{array}{ccc}3 x+4 y & 2 & x-2 y \\ a+b & 2 a-b & -1\end{array}\right]=\left[\begin{array}{ccc}2 & 2 & 4 \\ 5 & -5 & -1\end{array}\right]$

Sol :

Comparing the corresponding terms, we get.

3x + 4y = 2 ……(i)

x – 2y = 4 …….(ii)

Multiplying (i) by 1 and (ii) by 2

3x+4y=2

2 x-4 y=8

Adding we get, 5x=10

⇒x=2

$3 \times 2+4 y=2,6+4 y=2,4 y=2-6=-4$

y=-1

$\therefore x=2, y=-1$

a+b=5..(iii)

2a-b=-5...(iv)