1.
Write the order of the matrix $A = \begin{bmatrix} \sqrt{2} & 5 & -1 \\ 3 & 0 & 4 \\ 1 & 8 & \frac{1}{2} \end{bmatrix}$.
2.
If a matrix has $12$ elements, list all possible orders it can have.
3.
If a matrix has $5$ elements, what are the possible orders it can have?
4.
In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, find the elements $a_{13}, a_{21}, a_{33}, a_{24}$.
5.
Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are given by $a_{ij} = \frac{(i+j)^2}{2}$.
6.
Construct a $2 \times 3$ matrix whose elements are given by $a_{ij} = i - 2j$.
7.
Construct a $3 \times 2$ matrix whose elements are given by $a_{ij} = \frac{1}{2} |i - 3j|$.
8.
For a square matrix of order $n$, what is the relationship between the indices $i$ and $j$ for the elements lying on its principal diagonal?
9.
Classify the matrix $A = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ as diagonal, scalar, or identity matrix.
10.
Give an example of a row matrix which is also a column matrix.
11.
Find the trace of the matrix $B = \begin{bmatrix} -1 & 5 & 6 \\ 2 & 3 & 1 \\ 4 & -2 & 8 \end{bmatrix}$.
12.
If $A$ is a scalar matrix of order 3 and Trace$(A) = 15$, find the matrix $A$.
13.
Show that the matrix $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is an idempotent matrix.
14.
Verify if the matrix $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ is nilpotent. If yes, what is its index?
15.
A matrix $A$ is called involutory if $A^2 = I$. Verify if $A = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$ is involutory.
16.
What is the condition for a square matrix $A$ to be orthogonal?
17.
Find the values of $x, y$ and $z$ if $\begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\ 1 & 5 \end{bmatrix}$.
18.
Find $x, y, z$ from the equation: $\begin{bmatrix} x+y & 2 \\ 5+z & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$.
19.
Find the values of $a, b, c$ and $d$ from the equation: $\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}$.
20.
If $\begin{bmatrix} x+y+z \\ x+z \\ y+z \end{bmatrix} = \begin{bmatrix} 9 \\ 5 \\ 7 \end{bmatrix}$, find the values of $x, y,$ and $z$.
21.
Find the value of $x$ and $y$ if $2 \begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix} + \begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}$.
22.
Determine $x$ and $y$ such that $\begin{bmatrix} 2x-y \\ x+2y \end{bmatrix} = \begin{bmatrix} 5 \\ 5 \end{bmatrix}$.
23.
Are the matrices $A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ equal? Give a reason.
24.
Given $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$, find $A+B$.
25.
Given the same matrices $A$ and $B$ from Q24, find $A-B$.
26.
If $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$ and $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$, find $3A - C$.
27.
Compute the product: $\begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$.
28.
Compute the product: $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \begin{bmatrix} 2 & 3 & 4 \end{bmatrix}$.
29.
Compute the product: $\begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix}$.
30.
If $A = \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 2 \\ 4 \end{bmatrix}$, evaluate $AB$. Can you evaluate $BA$?
31.
Find matrix $X$ if $Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and $2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}$.
32.
If $X + Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$ and $X - Y = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$, find matrices $X$ and $Y$.
33.
Find the transpose of the matrix $A = \begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}$.
34.
Find the transpose of the matrix $B = \begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$.
35.
If $A = \begin{bmatrix} -1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix}$, verify that $(A+B)' = A' + B'$.
36.
If $A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, find $(A-B)'$.
37.
If $A = \begin{bmatrix} -2 \\ 4 \\ 5 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 3 & -6 \end{bmatrix}$, verify the reversal law: $(AB)' = B'A'$.
38.
If $A = \begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix}$, show that $(3A)' = 3A'$.
39.
If $A$ is a matrix of order $m \times n$, what is the order of $A'$?
40.
Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix.
41.
Show that the matrix $A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}$ is a skew-symmetric matrix.
42.
For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$, verify that $(A + A')$ is a symmetric matrix.
43.
For the same matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$, verify that $(A - A')$ is a skew-symmetric matrix.
44.
Find the values of $a$ and $b$ if the matrix $\begin{bmatrix} 0 & a \\ -3 & 0 \end{bmatrix}$ is a skew-symmetric matrix.
45.
Find the value of $x$ for which the matrix $A = \begin{bmatrix} 2 & x \\ 4 & 5 \end{bmatrix}$ is symmetric.
46.
Express the matrix $A = \begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
47.
If $A$ and $B$ are symmetric matrices of the same order, what can be said about $AB - BA$?
48.
Apply the elementary row operation $R_1 \leftrightarrow R_2$ on the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
49.
Apply the operation $C_2 \rightarrow C_2 - 2C_1$ on the matrix $B = \begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix}$.
50.
Apply the operation $R_2 \rightarrow R_2 - 3R_1$ to the matrix $\begin{bmatrix} 1 & 2 \\ 3 & 7 \end{bmatrix}$.
51.
If $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$, show that $AB = BA = I$.
52.
From Q51, what can we conclude about the relationship between matrix $A$ and matrix $B$?
53.
Write the initial matrix equation required to find the inverse of $A = \begin{bmatrix} 1 & 3 \\ 2 & 7 \end{bmatrix}$ using elementary row operations.
54.
Does the inverse of every square matrix exist? (Yes/No)
55.
If $A$ and $B$ are invertible matrices of the same order, simplify $(AB)^{-1}$.
