Types of Matrices - Practice Questions & MCQ

Equal Matrices: Two matrices are said to be equal if they have the same order and each element of one matrix is equal to the corresponding elements of another matrix or we can say $a_{i j}=b_{i j}$ where a is the element of one matrix and b is the element of another matrix.
Square matrix: the square matrix is the matrix in which the number of rows = number of columns. So a matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be a square matrix when m = n.

E.g.

$\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]_{3 \times 3}$ or, $\quad\left[\begin{array}{cc}2 & -4 \\ 7 & 3\end{array}\right]_{2 \times 2}$

Rectangular matrix: Rectangular matrix is the matrix in which is the number of rows ≠ and number of columns.

So a matrix $A=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be a rectangular matrix when $\mathrm{m} \neq \mathrm{n}$.
E.g.

$
\left[\begin{array}{llll}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34}
\end{array}\right]_{3 \times 4}
$

Null matrix/ Zero Matrix: A matrix whose all elements are 0, is called a null matrix. 

$\begin{aligned} & \mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}, \text { where } \mathrm{a}_{\mathrm{ij}}=0 \\ & \mathrm{Eg},\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]_{,}\left[\begin{array}{llll}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right]\end{aligned}$

Diagonal matrix: A square matrix is said to be a diagonal matrix, if all its elements except the diagonal elements are zero. 

So, a matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is a diagonal matrix if $\mathrm{a}_{\mathrm{ij}}=0$, whenever $\mathrm{i} \neq \mathrm{j}$ and $\mathrm{m}=\mathrm{n}$.
E.g

$
\left[\begin{array}{ccc}
a_{11} & 0 & 0 \\
0 & a_{22} & 0 \\
0 & 0 & a_{33}
\end{array}\right]
$

A diagonal matrix of order n x n having diagonal elements as d₁, d₂, d₃ ………, d_n is denoted by $\operatorname{diag}\left[d_1, d_2, d_3 \ldots \ldots \ldots, d_n\right]$

For example

$
A=\left[\begin{array}{cc}
6 & 0 \\
0 & -7
\end{array}\right] \quad B=\left[\begin{array}{ccc}
2 & 0 & 0 \\
0 & -9 & 0 \\
0 & 0 & 3
\end{array}\right]
$

so, we can write

$
\mathrm{A}=\operatorname{diag}[6,-7] \text { and } \mathrm{B}=\operatorname{diag}[2,-9,3]
$

Scalar matrix: A diagonal matrix whose all the diagonal elements are equal is called a scalar matrix.

$
\mathrm{A}=\left[\begin{array}{ll}
3 & 0 \\
0 & 3
\end{array}\right] \quad \mathrm{B}=\left[\begin{array}{ccc}
-3 & 0 & 0 \\
0 & -3 & 0 \\
0 & 0 & -3
\end{array}\right]
$
For a square matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n}}$ to be scalar matrix

$
\mathrm{a}_{\mathrm{ij}}= \begin{cases}0, & i \neq j \\ c, & i=j\end{cases}
$

Where c is not equal to 0 

Unit or Identity Matrix: A diagonal matrix of order n whose all the diagonal elements are equal to one is called an identity matrix of order n. It is represented as .

So, a square matrix $A=\left[a_{i j}\right]_{n \times n}$ is Identity matrix if

$
\mathrm{a}_{\mathrm{ij}}= \begin{cases}0, & i \neq j \\ 1, & i=j\end{cases}
$
For example,

$
\mathrm{I}_3=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
$