1.
Write the order of the matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$.
Answer: $2 \times 3$ (2 rows and 3 columns)
2.
If a matrix has $8$ elements, what are the possible orders it can have?
Answer: $1 \times 8, 8 \times 1, 2 \times 4, 4 \times 2$
3.
In the general representation of a matrix $A = [a_{ij}]$, what do the subscripts '$i$' and '$j$' represent?
Answer: '$i$' represents the row number and '$j$' represents the column number of the element.
4.
Identify the element $a_{23}$ in the matrix $A = \begin{bmatrix} 2 & -1 & 3 \\ 0 & 4 & 7 \end{bmatrix}$.
Answer: $7$ (Element in the 2nd row, 3rd column)
5.
Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are given by $a_{ij} = i + j$.
Answer: $A = \begin{bmatrix} (1+1) & (1+2) \\ (2+1) & (2+2) \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix}$
6.
A matrix having only one column is known as a _________________ matrix.
Answer: Column
7.
If the number of rows equals the number of columns, the matrix is called a _________________ matrix.
Answer: Square
8.
True or False: Every diagonal matrix must be a square matrix.
Answer: True
9.
Write a scalar matrix of order $2 \times 2$ with the diagonal scalar value $k=5$.
Answer: $\begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$
10.
Write the Identity (Unit) matrix of order $2$, denoted as $I_2$.
Answer: $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
11.
Write down the Zero (Null) matrix of order $2 \times 3$.
Answer: $\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$
12.
What is meant by the "Trace of a matrix"?
Answer: The sum of the elements lying along the principal diagonal of a square matrix.
13.
Find the trace of the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
Answer: Trace $(A) = 1 + 4 = 5$
14.
Fill in the blank: A square matrix $A$ is called idempotent if $A^2 =$ _________________.
Answer: $A$
15.
Fill in the blank: If a matrix $A$ satisfies $A^k = O$ for some positive integer $k$, it is called a _________________ matrix.
Answer: Nilpotent
16.
Fill in the blank: A matrix $A$ is involutory if $A^2 =$ _________________.
Answer: $I$ (Identity Matrix)
17.
Fill in the blank: A square matrix $A$ is orthogonal if $A A' = A' A =$ _________________.
Answer: $I$ (Identity Matrix)
18.
Two matrices $A$ and $B$ are equal if they have the same order and their _________________ elements are equal.
Answer: Corresponding
19.
Find the values of $x$ and $y$ if $\begin{bmatrix} x & 3 \\ 4 & y \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 4 & -2 \end{bmatrix}$.
Answer: $x = 1, y = -2$
20.
Can a matrix of order $2 \times 3$ ever be equal to a matrix of order $3 \times 2$? (Yes/No)
Answer: No (Their orders must be exactly identical).
21.
If $A = \begin{bmatrix} 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 4 \end{bmatrix}$, find the matrix addition $A + B$.
Answer: $\begin{bmatrix} 1+3 & 2+4 \end{bmatrix} = \begin{bmatrix} 4 & 6 \end{bmatrix}$
22.
State True or False: Matrix addition is commutative, meaning $A+B = B+A$.
Answer: True
23.
If $A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$, find the value of the scalar multiplication $\frac{1}{2} A$.
Answer: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
24.
For matrix multiplication $AB$ to be defined, the number of columns in $A$ must equal the number of _________________ in $B$.
Answer: Rows
25.
If matrix $A$ has order $3 \times 2$ and matrix $B$ has order $2 \times 4$, what will be the order of matrix $AB$?
Answer: $3 \times 4$
26.
Is matrix multiplication commutative in general? (Does $AB = BA$ always hold true?)
Answer: No
27.
Fill in the blank to complete the Associative Law for multiplication: $(AB)C =$ _________________.
Answer: $A(BC)$
28.
If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}$, write its transpose $A'$.
Answer: $A' = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix}$
29.
According to the properties of transpose, what is $(A')'$ equal to?
Answer: $A$
30.
State the Reversal Law for the transpose of a product: $(AB)' =$ _________________.
Answer: $B'A'$
31.
If $k$ is any scalar constant, what is $(kA)'$ equal to?
Answer: $k(A')$
32.
A square matrix $A$ is said to be symmetric if $A' =$ _________________.
Answer: $A$
33.
A square matrix $A$ is said to be skew-symmetric if $A' =$ _________________.
Answer: $-A$
34.
What must be the value of all principal diagonal elements in any skew-symmetric matrix?
Answer: Zero ($0$)
35.
Fill in the blanks: Any square matrix can be expressed as the sum of a _________________ matrix and a _________________ matrix.
Answer: Symmetric, Skew-Symmetric
36.
Write the formula used to express a square matrix $A$ as the sum of a symmetric and a skew-symmetric matrix.
Answer: $A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')$
37.
How many types of elementary row operations are there?
Answer: 3 types
38.
How many types of elementary column operations are there?
Answer: 3 types
39.
If there exists a matrix $B$ such that $AB = BA = I$, then $B$ is called the _________________ of $A$.
Answer: Inverse
40.
True or False: The inverse of a square matrix, if it exists, is unique.
Answer: True
41.
To calculate the inverse of matrix $A$ using elementary row operations, what equation do we write initially? ($A =$ _________)
Answer: $IA$
42.
To calculate the inverse of matrix $A$ using elementary column operations, what equation do we write initially? ($A =$ _________)
Answer: $AI$
