1.
Write the order of the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$.
2.
If a matrix has $8$ elements, what are the possible orders it can have?
3.
In the general representation of a matrix $A = [a_{ij}]$, what do the subscripts '$i$' and '$j$' represent?
4.
Identify the element $a_{23}$ in the matrix $A = \begin{bmatrix} 2 & -1 & 3 \\ 0 & 4 & 7 \end{bmatrix}$.
5.
Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are given by $a_{ij} = i + j$.
6.
A matrix having only one column is known as a _________________ matrix.
7.
If the number of rows equals the number of columns, the matrix is called a _________________ matrix.
8.
True or False: Every diagonal matrix must be a square matrix.
9.
Write a scalar matrix of order $2 \times 2$ with the diagonal scalar value $k=5$.
10.
Write the Identity (Unit) matrix of order $2$, denoted as $I_2$.
11.
Write down the Zero (Null) matrix of order $2 \times 3$.
12.
What is meant by the "Trace of a matrix"?
13.
Find the trace of the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
14.
Fill in the blank: A square matrix $A$ is called idempotent if $A^2 =$ _________________.
15.
Fill in the blank: If a matrix $A$ satisfies $A^k = O$ for some positive integer $k$, it is called a _________________ matrix.
16.
Fill in the blank: A matrix $A$ is involutory if $A^2 =$ _________________.
17.
Fill in the blank: A square matrix $A$ is orthogonal if $A A' = A' A =$ _________________.
21.
If $A = \begin{bmatrix} 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 4 \end{bmatrix}$, find the matrix addition $A + B$.
22.
State True or False: Matrix addition is commutative, meaning $A+B = B+A$.
23.
If $A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$, find the value of the scalar multiplication $\frac{1}{2} A$.
24.
For matrix multiplication $AB$ to be defined, the number of columns in $A$ must equal the number of _________________ in $B$.
25.
If matrix $A$ has order $3 \times 2$ and matrix $B$ has order $2 \times 4$, what will be the order of matrix $AB$?
26.
Is matrix multiplication commutative in general? (Does $AB = BA$ always hold true?)
27.
Fill in the blank to complete the Associative Law for multiplication: $(AB)C =$ _________________.
28.
If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$, write its transpose $A'$.
29.
According to the properties of transpose, what is $(A')'$ equal to?
30.
State the Reversal Law for the transpose of a product: $(AB)' =$ _________________.
31.
If $k$ is any scalar constant, what is $(kA)'$ equal to?
32.
A square matrix $A$ is said to be symmetric if $A' =$ _________________.
33.
A square matrix $A$ is said to be skew-symmetric if $A' =$ _________________.
34.
What must be the value of all principal diagonal elements in any skew-symmetric matrix?
35.
Fill in the blanks: Any square matrix can be expressed as the sum of a _________________ matrix and a _________________ matrix.
36.
Write the formula used to express a square matrix $A$ as the sum of a symmetric and a skew-symmetric matrix.
37.
How many types of elementary row operations are there?
38.
How many types of elementary column operations are there?
39.
If there exists a matrix $B$ such that $AB = BA = I$, then $B$ is called the _________________ of $A$.
40.
True or False: The inverse of a square matrix, if it exists, is unique.
41.
To calculate the inverse of matrix $A$ using elementary row operations, what equation do we write initially? ($A =$ _________)
42.
To calculate the inverse of matrix $A$ using elementary column operations, what equation do we write initially? ($A =$ _________)
