1.
State the order of the matrix $A = \begin{bmatrix} 4 & -1 \\ 0 & 3 \\ 5 & 2
\end{bmatrix}$.
2.
If a matrix has $8$ elements, what are the possible orders it can have?
3.
Write an identity matrix of order $2 \times 2$ and denote it by $I$.
4.
Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are given by
$a_{ij} = 2i - j$.
5.
Construct a $2 \times 3$ matrix whose elements are given by $a_{ij} = |i - 2j|$.
6.
Classify the following matrices as row, column, square, or zero matrix: (i)
$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ (ii) $\begin{bmatrix} 3 & -1 & 4 \end{bmatrix}$ (iii)
$\begin{bmatrix} -5 \\ 2 \end{bmatrix}$.
7.
Find the values of $x$ and $y$ if $\begin{bmatrix} 2x + y \\ x - 3y \end{bmatrix}
= \begin{bmatrix} 5 \\ -1 \end{bmatrix}$.
8.
Find the values of $a, b, c$ and $d$ if $\begin{bmatrix} a - b & 2a + c \\ 2a - b
& 3c + d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$.
9.
Given $\begin{bmatrix} x^2 \\ y^2 \end{bmatrix} = \begin{bmatrix} 9 \\ 16
\end{bmatrix}$ and $x < 0, y> 0$, find the values of $x$ and $y$.
10.
Find $x$ and $y$ from the given equation: $\begin{bmatrix} 2x - 3 & y + 4 \\ 0 &
5 \end{bmatrix} = \begin{bmatrix} 7 & 6 \\ 0 & 5 \end{bmatrix}$.
11.
Determine whether the matrices $A = \begin{bmatrix} 1 & 2 \end{bmatrix}$ and $B =
\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ are equal. Give reasons.
12.
If $\begin{bmatrix} x & y - z \\ z + x & a \end{bmatrix} = \begin{bmatrix} 3 & -1
\\ 5 & 2 \end{bmatrix}$, find the values of $x, y, z$ and $a$.
13.
For what values of $x$ and $y$ are the following matrices equal? $A =
\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}$, $B = \begin{bmatrix} 0 & y-2 \\ 8 & 4
\end{bmatrix}$.
14.
If matrix $\begin{bmatrix} p^2 - 1 & 0 \\ 2 & q^2 \end{bmatrix} = \begin{bmatrix}
3 & 0 \\ 2 & 4 \end{bmatrix}$ and $p>0, q<0$, find $p$ and $q$.
15.
Define a principal diagonal in a square matrix. What are the diagonal
elements in $M = \begin{bmatrix} 5 & -2 \\ 3 & 7 \end{bmatrix}$?
16.
Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$ and $B =
\begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$. Find $A + B$.
17.
Using matrices from Q16, compute $A - B$.
18.
Using matrices from Q16, compute $2A + 3B$.
19.
If $P = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$ and $Q =
\begin{bmatrix} 4 & 1 \\ 2 & 0 \end{bmatrix}$, find a matrix $R$ such that $P + R = Q$.
20.
Find the additive inverse of matrix $M = \begin{bmatrix} -5 & 6 \\ 2 & -8
\end{bmatrix}$.
21.
Given $A = \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix}$ and $B =
\begin{bmatrix} -1 & 4 \\ 2 & 1 \end{bmatrix}$, evaluate $3A - 2B$.
22.
If $A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix}$ and $B =
\begin{bmatrix} -4 & -1 \\ -3 & -2 \end{bmatrix}$, find $X$ such that $2A + X = B$.
23.
Find $x$ and $y$ if $2 \begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} +
\begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}$.
24.
If $\begin{bmatrix} 2 & -3 \\ 1 & 4 \end{bmatrix} + X = \begin{bmatrix} 0 & 0
\\ 0 & 0 \end{bmatrix}$, find matrix $X$.
25.
Given $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $I =
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Show that $A + A = 2A$.
26.
If $X - \begin{bmatrix} 1 & -2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} -1 &
5 \\ 0 & -2 \end{bmatrix}$, find matrix $X$.
27.
Solve for $a, b, c, d$: $3 \begin{bmatrix} a & b \\ c & d \end{bmatrix} =
\begin{bmatrix} a & 6 \\ -1 & 2d \end{bmatrix} + \begin{bmatrix} 4 & a+b \\ c+d & 3
\end{bmatrix}$.
28.
If $A = \begin{bmatrix} 2 & 5 \\ -1 & 4 \end{bmatrix}$ and $B =
\begin{bmatrix} -1 & -4 \\ 2 & 3 \end{bmatrix}$, find $C$ such that $A + B + C$ is a zero
matrix.
29.
Simplify: $\cos \theta \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin
\theta & \cos \theta \end{bmatrix} + \sin \theta \begin{bmatrix} \sin \theta & -\cos \theta \\
\cos \theta & \sin \theta \end{bmatrix}$.
30.
Given $A = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$ and $B =
\begin{bmatrix} -4 \\ 2 \\ 5 \end{bmatrix}$, evaluate $\frac{1}{2}A - B$.
31.
If matrix $A$ is of order $3 \times 2$ and $B$ is of order $2 \times 4$, what
is the order of matrix $AB$? Can we find $BA$?
32.
Given $A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$ and $B =
\begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix}$, find $AB$.
33.
Using the matrices from Q32, find $BA$. Is $AB = BA$?
34.
Evaluate: $\begin{bmatrix} 3 & -2 \\ -1 & 4 \end{bmatrix} \begin{bmatrix} 2
\\ 5 \end{bmatrix}$.
35.
Evaluate: $\begin{bmatrix} 1 & -2 \end{bmatrix} \begin{bmatrix} 3 \\ 4
\end{bmatrix}$.
36.
If $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, prove that $A^2 = I$
where $I$ is the identity matrix of order $2 \times 2$.
37.
Given $A = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}$, find $A^2$.
38.
If $A = \begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}$ and $B =
\begin{bmatrix} 4 & 0 \\ 1 & 5 \end{bmatrix}$, find $(AB)^T$ (where $T$ denotes transpose -
converting rows to columns).
39.
Find the value of $x$ if $\begin{bmatrix} x & 1 \end{bmatrix} \begin{bmatrix}
1 & 0 \\ -2 & -3 \end{bmatrix} \begin{bmatrix} x \\ 5 \end{bmatrix} = 0$.
40.
If $A = \begin{bmatrix} 2 & x \\ y & 3 \end{bmatrix}$ and $A \begin{bmatrix}
1 \\ 2 \end{bmatrix} = \begin{bmatrix} 4 \\ 7 \end{bmatrix}$, find $x$ and $y$.
41.
Let $A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$ and $B =
\begin{bmatrix} 2 & 3 \\ -1 & 0 \end{bmatrix}$. Verify that $A(A + B) = A^2 + AB$.
42.
Evaluate: $\begin{bmatrix} 4 \sin 30^\circ & \cos 60^\circ \\ \cos 90^\circ &
2 \sin 45^\circ \end{bmatrix} \begin{bmatrix} 2 \\ \sqrt{2} \end{bmatrix}$.
43.
If $A = \begin{bmatrix} a & 0 \\ 0 & a \end{bmatrix}$ and $B =
\begin{bmatrix} x & y \\ z & w \end{bmatrix}$, prove that $AB = aB$.
44.
If $M \times \begin{bmatrix} 1 & 1 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} 1
& 2 \end{bmatrix}$, find the order of matrix $M$ and the matrix $M$.
45.
Find $x$ and $y$ if $\begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 18 \end{bmatrix}$.
46.
Given $A = \begin{bmatrix} 3 & -2 \\ -1 & 4 \end{bmatrix}$, $B =
\begin{bmatrix} 6 \\ 1 \end{bmatrix}$, $C = \begin{bmatrix} -4 \\ 5 \end{bmatrix}$ and $D =
\begin{bmatrix} 2 \\ 2 \end{bmatrix}$. Find $AB + 2C - 4D$.
47.
Find the product $\begin{bmatrix} a & b \\ -b & a \end{bmatrix}
\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$.
48.
If $A = \begin{bmatrix} 2 & 0 \\ -1 & 7 \end{bmatrix}$, find $k$ so that $A^2
= 8A + kI$.
49.
Evaluate: $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} a &
b \\ c & d \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.
50.
Let $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$. Find $f(A)$ where
$f(x) = x^2 - 5x + 7$.
51.
Find the matrix $X$ if $X + \begin{bmatrix} 2 & 5 \\ 3 & 2 \end{bmatrix} =
\begin{bmatrix} 4 & 0 \\ -7 & 6 \end{bmatrix}$.
52.
If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B =
\begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}$, find matrix $X$ such that $2A + X = B$.
53.
Determine the order of matrix $X$ and find $X$ given that: $\begin{bmatrix} 2
& 1 \\ 0 & -2 \end{bmatrix} X = \begin{bmatrix} 4 & 1 \\ 6 & -8 \end{bmatrix}$.
54.
Find the matrix $X$ such that $\begin{bmatrix} 1 & 2 \\ -1 & 3 \end{bmatrix}
X = \begin{bmatrix} 5 \\ 0 \end{bmatrix}$.
55.
Given $\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y
\end{bmatrix} = \begin{bmatrix} 5 \\ 3 \end{bmatrix}$, solve for $x$ and $y$.
56.
If $A = \begin{bmatrix} -1 & 0 \\ 2 & 3 \end{bmatrix}$ and $I =
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, find matrix $X$ if $A^2 - 2A + X = I$.
57.
Solve the matrix equation: $X \begin{bmatrix} -1 \\ 2 \end{bmatrix} =
\begin{bmatrix} 3 \\ 4 \end{bmatrix}$. (Identify order of X first).
58.
Let $A = \begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix}$ and $B =
\begin{bmatrix} 2 & 1 \\ 4 & -1 \end{bmatrix}$. Find a matrix $C$ such that $2A - 3B + C = 0$.
59.
Given $A = \begin{bmatrix} 1 & 1 \\ x & y \end{bmatrix}$ and $A^2 = A$, find
$x$ and $y$.
60.
Find $x$ and $y$ if $\begin{bmatrix} x & y \\ 1 & 2 \end{bmatrix}
\begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} 5 \\ 0 \end{bmatrix}$.
61.
(ICSE) Given $A = \begin{bmatrix} 2 & -1 \\ 2 & 0
\end{bmatrix}$, $B = \begin{bmatrix} -3 & 2 \\ 4 & 0 \end{bmatrix}$ and $C = \begin{bmatrix} 1 &
0 \\ 0 & 2 \end{bmatrix}$, find the matrix $X$ such that $A + X = 2B + C$.
62.
(ICSE) Find $x$ and $y$, if $\begin{bmatrix} 2x & x \\ y &
3y \end{bmatrix} \begin{bmatrix} 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 16 \\ 9 \end{bmatrix}$.
63.
(ICSE) Let $A = \begin{bmatrix} 4 & -2 \\ 6 & -3
\end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix}$. Find $A^2 - A + B$.
64.
(ICSE) If $A = \begin{bmatrix} 1 & 3 \\ 2 & 4
\end{bmatrix}$, $B = \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix}$, and $C = \begin{bmatrix} 4 &
3 \\ 1 & 2 \end{bmatrix}$. Find a matrix $D$ such that $A(B + C) - D = 0$.
65.
(ICSE) Given $A = \begin{bmatrix} \sec \theta & -\tan \theta
\\ \tan \theta & \sec \theta \end{bmatrix}$ and $B = \begin{bmatrix} \sec \theta & \tan \theta
\\ -\tan \theta & \sec \theta \end{bmatrix}$. Prove that $AB = I$.
66.
(ICSE) Evaluate without using tables: $\begin{bmatrix} 2
\cos 60^\circ & -2 \sin 30^\circ \\ -\tan 45^\circ & \cos 0^\circ \end{bmatrix} \begin{bmatrix}
\cot 45^\circ & \text{cosec } 30^\circ \\ \sec 60^\circ & \sin 90^\circ \end{bmatrix}$.
67.
(ICSE) If $\begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix} +
2M = 3 \begin{bmatrix} 3 & 2 \\ 0 & -3 \end{bmatrix}$, find matrix $M$.
68.
(ICSE) A store sells items A and B. The quantity of A sold
is $x$ and B is $y$, represented by matrix $Q = \begin{bmatrix} x \\ y \end{bmatrix}$. The
prices are ?10 and ?15 respectively, represented by $P = \begin{bmatrix} 10 & 15 \end{bmatrix}$.
If the total revenue $PQ = [350]$, and $x+y=30$, frame matrix equations and solve for $x$ and
$y$.
69.
(ICSE) Determine the matrix $X$ in the equation:
$\begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} X = \begin{bmatrix} 2 & 1 \\ 1 & 1
\end{bmatrix}$.
70.
(ICSE) If $A = \begin{bmatrix} 3 & a \\ -1 & b
\end{bmatrix}$ and $A^2 = I$, find the values of $a$ and $b$.
