## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3

These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3

More Exercises

Question 1.
If A = $$\begin{bmatrix} 3 & \quad 5 \\ 4 & \quad -2 \end{bmatrix}$$ and B = $$\left[ \begin{matrix} 2 \\ 4 \end{matrix} \right]$$, is the product AB possible ? Give a reason. If yes, find AB.
Solution:
Yes, the product is possible because of
number of column in A = number of row in B
i.e., (2 x 2). (2 x 1) = (2 x 1) is the order of the matrix.

Question 2.
If A = $$\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$$,B = $$\begin{bmatrix} 1 & -1 \\ -3 & 2 \end{bmatrix}$$, find AB and BA, Is AB = BA ?
Solution:
A = $$\begin{bmatrix} 2 & 5 \\ 1 & 3 \end{bmatrix}$$,
B = $$\begin{bmatrix} 1 & -1 \\ -3 & 2 \end{bmatrix}$$

Question 3.
If P = $$\begin{bmatrix} 4 & 6 \\ 2 & -8 \end{bmatrix}$$,Q = $$\begin{bmatrix} 2 & -3 \\ -1 & 1 \end{bmatrix}$$
Find 2PQ
Solution:
P = $$\begin{bmatrix} 4 & 6 \\ 2 & -8 \end{bmatrix}$$,
Q = $$\begin{bmatrix} 2 & -3 \\ -1 & 1 \end{bmatrix}$$
$$2PQ=2\begin{bmatrix} 4 & \quad 6 \\ 2 & -8 \end{bmatrix}\times \begin{bmatrix} 2\quad & -3 \\ -1 & \quad 1 \end{bmatrix}$$

Question 4.
Given A = $$\begin{bmatrix} 1 & 1 \\ 8 & 3 \end{bmatrix}$$ , evaluate A² – 4A
Solution:
A = $$\begin{bmatrix} 1 & 1 \\ 8 & 3 \end{bmatrix}$$
A² – 4A = $$\begin{bmatrix} 1 & \quad 1 \\ 8 & \quad 3 \end{bmatrix}\begin{bmatrix} 1\quad & 1 \\ 8\quad & 3 \end{bmatrix}-4\begin{bmatrix} 1\quad & 1 \\ 8\quad & 3 \end{bmatrix}$$

Question 5.
If A = $$\begin{bmatrix} 3 & \quad 7 \\ 2 & \quad 4 \end{bmatrix}$$, B = $$\begin{bmatrix} 0 & \quad 2 \\ 5 & \quad 3 \end{bmatrix}$$ and C = $$\begin{bmatrix} 1 & \quad -5 \\ -4 & \quad 6 \end{bmatrix}$$
Find AB – 5C
Solution:
A = $$\begin{bmatrix} 3 & \quad 7 \\ 2 & \quad 4 \end{bmatrix}$$, B = $$\begin{bmatrix} 0 & \quad 2 \\ 5 & \quad 3 \end{bmatrix}$$ and C = $$\begin{bmatrix} 1 & \quad -5 \\ -4 & \quad 6 \end{bmatrix}$$
AB = $$\begin{bmatrix} 3 & \quad 7 \\ 2 & \quad 4 \end{bmatrix}$$$$\begin{bmatrix} 0 & \quad 2 \\ 5 & \quad 3 \end{bmatrix}$$

Question 6.
If A = $$\begin{bmatrix} 1 & \quad 2 \\ 2 & \quad 1 \end{bmatrix}$$ and B = $$\begin{bmatrix} 2 & \quad 1 \\ 1 & \quad 2 \end{bmatrix}$$, find A(BA)
Solution:
A = $$\begin{bmatrix} 1 & \quad 2 \\ 2 & \quad 1 \end{bmatrix}$$
B = $$\begin{bmatrix} 2 & \quad 1 \\ 1 & \quad 2 \end{bmatrix}$$

Question 7.
Given matrices:
A = $$\begin{bmatrix} 2 & \quad 1 \\ 4 & \quad 2 \end{bmatrix}$$ and B = $$\begin{bmatrix} 3 & \quad 4 \\ -1 & \quad -2 \end{bmatrix}$$, C = $$\begin{bmatrix} -3 & \quad 1 \\ 0 & \quad -2 \end{bmatrix}$$
Find the products of (i) ABC (ii) ACB and state whether they are equal.
Solution:
A = $$\begin{bmatrix} 2 & \quad 1 \\ 4 & \quad 2 \end{bmatrix}$$
B = $$\begin{bmatrix} 3 & \quad 4 \\ -1 & \quad -2 \end{bmatrix}$$,
C = $$\begin{bmatrix} -3 & \quad 1 \\ 0 & \quad -2 \end{bmatrix}$$

Question 8.
Evaluate : $$\begin{bmatrix} 4\sin { { 30 }^{ o } } & \quad 2cos{ 60 }^{ o } \\ sin{ 90 }^{ o } & \quad 2cos{ 0 }^{ o } \end{bmatrix}\begin{bmatrix} 4 & 5 \\ 5 & 4 \end{bmatrix}$$
Solution:
$$\begin{bmatrix} 4\sin { { 30 }^{ o } } & \quad 2cos{ 60 }^{ o } \\ sin{ 90 }^{ o } & \quad 2cos{ 0 }^{ o } \end{bmatrix}\begin{bmatrix} 4 & 5 \\ 5 & 4 \end{bmatrix}$$
$$sin{ 30 }^{ o }=\frac { 1 }{ 2 } ,cos{ 60 }^{ o }=\frac { 1 }{ 2 }$$

Question 9.
If A = $$\begin{bmatrix} -1 & \quad 3 \\ 2 & \quad 4 \end{bmatrix}$$, B = $$\begin{bmatrix} 2 & \quad -3 \\ -4 & \quad -6 \end{bmatrix}$$ find the matrix AB + BA
Solution:
A = $$\begin{bmatrix} -1 & \quad 3 \\ 2 & \quad 4 \end{bmatrix}$$,
B = $$\begin{bmatrix} 2 & \quad -3 \\ -4 & \quad -6 \end{bmatrix}$$
$$AB=\begin{bmatrix} -1 & \quad 3 \\ 2 & \quad 4 \end{bmatrix}\times \begin{bmatrix} 2 & \quad -3 \\ -4 & \quad -6 \end{bmatrix}$$

Question 10.
A = $$\begin{bmatrix} 1 & \quad 2 \\ 3 & \quad 4 \end{bmatrix}$$ and B = $$\begin{bmatrix} 6 & \quad 1 \\ 1 & \quad 1 \end{bmatrix}$$, C = $$\begin{bmatrix} -2 & \quad -3 \\ 0 & \quad 1 \end{bmatrix}$$
find each of the following and state if they are equal.
(i) CA + B
(ii) A + CB
Solution:
(i) CA + B
CA = $$\begin{bmatrix} -2 & \quad -3 \\ 0 & \quad 1 \end{bmatrix}$$$$\begin{bmatrix} 1 & \quad 2 \\ 3 & \quad 4 \end{bmatrix}$$

Question 11.
If A = $$\begin{bmatrix} 1 & -2 \\ 2 & -1 \end{bmatrix}$$ and B = $$\begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}$$
Find 2B – A²
Solution:
A = $$\begin{bmatrix} 1 & -2 \\ 2 & -1 \end{bmatrix}$$
B = $$\begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}$$
2B = $$2\begin{bmatrix} 3 & 2 \\ -2 & 1 \end{bmatrix}$$
= $$\begin{bmatrix} 6 & 4 \\ -4 & 2 \end{bmatrix}$$

Question 12.
If A = $$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ and B = $$\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}$$, C = $$\begin{bmatrix} 5 & 1 \\ 7 & 4 \end{bmatrix}$$, compute
(i) A(B + C)
(ii) (B + C)A
Solution:
(i) A(B + C)
A = $$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$
B = $$\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}$$,
C = $$\begin{bmatrix} 5 & 1 \\ 7 & 4 \end{bmatrix}$$

Question 13.
If A = $$\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$$ and B = $$\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}$$, C = $$\begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix}$$
find the matrix C(B – A)
Solution:
A = $$\begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$$
B = $$\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}$$,
C = $$\begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix}$$

Question 14.
A = $$\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$$ and B = $$\begin{bmatrix} 2 & 3 \\ -1 & 0 \end{bmatrix}$$
Find A² + AB + B²
Solution:
Given that
A = $$\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$$
B = $$\begin{bmatrix} 2 & 3 \\ -1 & 0 \end{bmatrix}$$

Question 15.
If A = $$\begin{bmatrix} 2 & 1 \\ 0 & -2 \end{bmatrix}$$ and B = $$\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}$$, C = $$\begin{bmatrix} -3 & 2 \\ -1 & 4 \end{bmatrix}$$
Find A² + AC – 5B
Solution:
A = $$\begin{bmatrix} 2 & 1 \\ 0 & -2 \end{bmatrix}$$
B = $$\begin{bmatrix} 4 & 1 \\ -3 & -2 \end{bmatrix}$$,
C = $$\begin{bmatrix} -3 & 2 \\ -1 & 4 \end{bmatrix}$$

Question 16.
If A = $$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$, find A2 and A3.Also state that which of these is equal to A
Solution:
A = $$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$
A² = A x A = $$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$$$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$$

Question 17.
If X = $$\begin{bmatrix} 4 & 1 \\ -1 & 2 \end{bmatrix}$$, show that 6X – X² = 9I Where I is the unit matrix.
Solution:
Given that
X = $$\begin{bmatrix} 4 & 1 \\ -1 & 2 \end{bmatrix}$$

Question 18.
Show that $$\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$$ is a solution of the matrix equation X² – 2X – 3I = 0,Where I is the unit matrix of order 2
Solution:
Given
X² – 2X – 3I = 0
Solution = $$\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$$
or
X = $$\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$$
∴ X² = $$\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$$$$\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$$

Question 19.
Find the matrix X of order 2 × 2 which satisfies the equation
$$\begin{bmatrix} 3 & 7 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} 0 & 2 \\ 5 & 3 \end{bmatrix}+2X=\begin{bmatrix} 1 & -5 \\ -4 & 6 \end{bmatrix}$$
Solution:
Given
$$\begin{bmatrix} 3 & 7 \\ 2 & 4 \end{bmatrix}\begin{bmatrix} 0 & 2 \\ 5 & 3 \end{bmatrix}+2X=\begin{bmatrix} 1 & -5 \\ -4 & 6 \end{bmatrix}$$

Question 20.
If A = $$\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}$$, find the value of x, so that A² – 0
Solution:
Given
A = $$\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}$$
A² = $$\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}$$$$\begin{bmatrix} 1 & 1 \\ x & x \end{bmatrix}$$

Question 21.
If $$\begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} x \\ 0 \end{matrix} \right]$$ Find the value of x
Solution:
$$\begin{bmatrix} 1 & 3 \\ 0 & 0 \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} x \\ 0 \end{matrix} \right]$$
⇒ $$\begin{bmatrix} 2 & -3 \\ 0 & 0 \end{bmatrix}=\left[ \begin{matrix} x \\ 0 \end{matrix} \right]$$
⇒ $$\left[ \begin{matrix} -1 \\ 0 \end{matrix} \right] =\left[ \begin{matrix} x \\ 0 \end{matrix} \right]$$
Comparing the corresponding elements
x = -1

Question 22.
(i) Find x and y if $$\begin{bmatrix} -3 & 2 \\ 0 & -5 \end{bmatrix}\left[ \begin{matrix} x \\ 2 \end{matrix} \right] =\left[ \begin{matrix} -5 \\ y \end{matrix} \right]$$
(ii) Find x and y if $$\begin{bmatrix} 2x & x \\ y & 3y \end{bmatrix}\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right] =\left[ \begin{matrix} 16 \\ 9 \end{matrix} \right]$$
Solution:
(i) $$\begin{bmatrix} -3 & 2 \\ 0 & -5 \end{bmatrix}\left[ \begin{matrix} x \\ 2 \end{matrix} \right] =\left[ \begin{matrix} -5 \\ y \end{matrix} \right]$$
⇒ $$\begin{bmatrix} -3x & 4 \\ 0 & -10 \end{bmatrix}=\left[ \begin{matrix} -5 \\ y \end{matrix} \right]$$

Here x = 2, y = 1

Question 23.
Find x and y if
$$\begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right]$$
Solution:
Given
$$\begin{bmatrix} x+y & y \\ 2x & x-y \end{bmatrix}\left[ \begin{matrix} 2 \\ -1 \end{matrix} \right] =\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right]$$

Question 24.
If $$\begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}\begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}=\begin{bmatrix} x & 0 \\ 9 & 0 \end{bmatrix}$$ find the values of x and y
Solution:
Given
$$\begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix}\begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}=\begin{bmatrix} x & 0 \\ 9 & 0 \end{bmatrix}$$

Question 25.
If $$\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ write down the values of a,b,c and d
Solution:
Given
$$\begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

Comparing the corresponding elements
a = 3, b = 4, c = 2, d = 5

Question 26.
Find the value of x given that A² = B
Where A = $$\begin{bmatrix} 2 & 12 \\ 0 & 1 \end{bmatrix}$$ and
B = $$\begin{bmatrix} 4 & x \\ 0 & 1 \end{bmatrix}$$
Solution:
A = $$\begin{bmatrix} 2 & 12 \\ 0 & 1 \end{bmatrix}$$ and
B = $$\begin{bmatrix} 4 & x \\ 0 & 1 \end{bmatrix}$$
A² = B

Question 27.
If A = $$\begin{bmatrix} 2 & x \\ 0 & 1 \end{bmatrix}$$ and B = $$\begin{bmatrix} 4 & 36 \\ 0 & 1 \end{bmatrix}$$, find the value of x, given that A² – B
Solution:
Given
A² = $$\begin{bmatrix} 2 & x \\ 0 & 1 \end{bmatrix}$$$$\begin{bmatrix} 2 & x \\ 0 & 1 \end{bmatrix}$$

Question 28.
If A = $$\begin{bmatrix} 3 & x \\ 0 & 1 \end{bmatrix}$$ and B = $$\begin{bmatrix} 9 & 16 \\ 0 & -y \end{bmatrix}$$ find x and y when A² = B
Solution:
Given
A = $$\begin{bmatrix} 3 & x \\ 0 & 1 \end{bmatrix}$$ and B = $$\begin{bmatrix} 9 & 16 \\ 0 & -y \end{bmatrix}$$ find x and y when A² = B

Question 29.
Find x, y if $$\begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix}\left[ \begin{matrix} -1 \\ 2x \end{matrix} \right] +3\left[ \begin{matrix} -2 \\ 1 \end{matrix} \right] =2\left[ \begin{matrix} y \\ 3 \end{matrix} \right]$$
Solution:
Given
$$\begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix}\left[ \begin{matrix} -1 \\ 2x \end{matrix} \right] +3\left[ \begin{matrix} -2 \\ 1 \end{matrix} \right] =2\left[ \begin{matrix} y \\ 3 \end{matrix} \right]$$

Question 30.
If $$\begin{bmatrix} a & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 4 & 3 \\ -3 & 2 \end{bmatrix}=\begin{bmatrix} b & 11 \\ 4 & c \end{bmatrix}$$ find a,b and c
Solution:
$$\begin{bmatrix} a & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 4 & 3 \\ -3 & 2 \end{bmatrix}=\begin{bmatrix} b & 11 \\ 4 & c \end{bmatrix}$$
⇒ $$\begin{bmatrix} 4a-3 & 3a+2 \\ 4+0 & 3+0 \end{bmatrix}=\begin{bmatrix} b & 11 \\ 4 & c \end{bmatrix}$$

Question 31.
If A = $$\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}$$ ,B = $$\begin{bmatrix} 2 & x \\ 0 & -\frac { 1 }{ 2 } \end{bmatrix}$$ find the value of x if AB = BA
Solution:
Given
AB = $$\begin{bmatrix} 1 & 4 \\ 0 & -1 \end{bmatrix}$$$$\begin{bmatrix} 2 & x \\ 0 & -\frac { 1 }{ 2 } \end{bmatrix}$$

Question 32.
If A = $$\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$$ find x and y so that A² – xA + yI
Solution:
Given
A² = $$\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$$$$\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$$

Question 33.
If P = $$\begin{bmatrix} 2 & 6 \\ 3 & 9 \end{bmatrix}$$, Q = $$\begin{bmatrix} 3 & x \\ y & 2 \end{bmatrix}$$
find x and y such that PQ = 0
Solution:
Given
P = $$\begin{bmatrix} 2 & 6 \\ 3 & 9 \end{bmatrix}$$,
Q = $$\begin{bmatrix} 3 & x \\ y & 2 \end{bmatrix}$$

Question 34.
Let $$M\times \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix}=\left[ \begin{matrix} 1 & 2 \end{matrix} \right]$$ where M is a matrix
(i) State the order of matrix M
(ii) Find the matrix M
Solution:
Given
(i) M is the order of 1 x 2
let M = [x y]

Question 35.
Given $$\begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix}$$ ,X = $$\left[ \begin{matrix} 7 \\ 6 \end{matrix} \right]$$
(i) the order of the matrix X
(ii) the matrix X
Solution:
We have
$$\begin{bmatrix} 2 & 1 \\ -3 & 4 \end{bmatrix}$$ , X = $$\left[ \begin{matrix} 7 \\ 6 \end{matrix} \right]$$

Question 36.
Solve the matrix equation : $$\left[ \begin{matrix} 4 \\ 1 \end{matrix} \right]$$ ,X = $$\begin{bmatrix} -4 & 8 \\ -1 & 2 \end{bmatrix}$$
Solution:
$$\left[ \begin{matrix} 4 \\ 1 \end{matrix} \right]$$ , X = $$\begin{bmatrix} -4 & 8 \\ -1 & 2 \end{bmatrix}$$
Let matrix X = [x y]

Question 37.
(i) If A = $$\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}$$ and B = $$\left[ \begin{matrix} -3 \\ 2 \end{matrix} \right]$$ find the matrix C such that AC = B
(ii) If A = $$\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}$$ and B = [0 -3] find the matrix C such that CA = B
Solution:
(i) given
A = $$\begin{bmatrix} 2 & -1 \\ -4 & 5 \end{bmatrix}$$
B = $$\left[ \begin{matrix} -3 \\ 2 \end{matrix} \right]$$

Question 38.
If A = $$\begin{bmatrix} 3 & -4 \\ -1 & 2 \end{bmatrix}$$ , find matrix B such that BA = I,where I is unity matrix of order 2
Solution:
A = $$\begin{bmatrix} 3 & -4 \\ -1 & 2 \end{bmatrix}$$
BA = I, where I is unity matrix of order 2

Question 39.
If B = $$\begin{bmatrix} -4 & 2 \\ 5 & -1 \end{bmatrix}$$ and C = $$\begin{bmatrix} 17 & -1 \\ 47 & -13 \end{bmatrix}$$
find the matrix A such that AB = C
Solution:
B = $$\begin{bmatrix} -4 & 2 \\ 5 & -1 \end{bmatrix}$$
C = $$\begin{bmatrix} 17 & -1 \\ 47 & -13 \end{bmatrix}$$
and AB = C

Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 8 Matrices Ex 8.3 are helpful to complete your math homework.

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