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

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

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

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:

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:

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

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

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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

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:

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:

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

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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:

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

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:

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:

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:

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

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:

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:

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:

ML Aggarwal Class 10 Solutions for ICSE Maths