Types of Matrices

A matrix is a rectangular arrangement of symbols along rows and columns that might be real or complex numbers. Thus, a system of m x n symbols arranged in a rectangular formation along m rows and n columns and bonded by the brackets [ ] is called an m by n matrix (which is written as m x n matrix).

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

In a compact form, the above matrix is represented by $A=\left[a_{i j}\right]$

The numbers $a_{11}$, $a_{12}$ etc. are called the elements of the matrix.

The element $\begin{equation}
a_{i j}
\end{equation}$ belongs to the ith row and jth column and is called the (i,j) th element of the matrix.

In this article, we will cover the concept of Types of Matrices. This category falls under the broader category of Matrices, which is a crucial Chapter in class 11 Mathematics. It is essential not only for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more.

Types of Matrices

Matrices are distinguished based on their order elements and certain other conditions. There are different types of matrices but the most commonly used are discussed below. Let’s find out the types of matrices in the field of mathematics.

Row matrix: A matrix containing only one row is called a row matrix. So a matrix $A=\left[a_{i j}\right]_{m \times n}$ is said to be a row matrix when $m=1$.

It can be denoted by

$
\left[\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & \ldots & \ldots & a_{1 n}
\end{array}\right]_{1 \times \mathrm{n}}
$

For example, $\left[\begin{array}{llll}1 & 32 & 81 & -32\end{array}\right]$ has only 1 row. It has order $1 \times 4$

Column matrix: A matrix containing only one column is known as a column matrix. So a matrix $A=\left[a_{i j}\right]_m \times n$ is said to be a column matrix when $n=1$. It is denoted by:-

$\left[\begin{array}{c}a_{11} \\ a_{21} \\ a_{31} \\ \cdots \\ \cdots \\ a_{m 1}\end{array}\right]_{\mathrm{m} \times 1}$

For example:-

$\left[\begin{array}{c}2 \\ 32 \\ 3 \\ 7\end{array}\right]$

This matrix has order 4 x 1

Note: A matrix that contains only one row or one column is also known as a vector i.e. row vectors and column vectors.

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 $\mathrm{a}$ is the element of one matrix and $\mathrm{b}$ 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 $A=\left[a_{i j}\right]_m \times n$ is said to be a square matrix when $\mathrm{m}=\mathrm{n}$.
Example,
$
\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} \text { or, }\left[\begin{array}{cc}
2 & -4 \\
7 & 3
\end{array}\right]_{2 \times 2}
$
$A+A^{\prime}$ is a symmetric matrix and $A-A^{\prime}$ is a skew-symmetric matrix for a square matrix with real number elements.
A symmetric and skew-symmetric matrix are added together to yield a square matrix. $1 / 2\left(A+A^{\prime}\right)+1 / 2\left(A-A^{\prime}\right)$ equals $A$.

Rectangular matrix: Rectangular matrix is the matrix in which is the number of rows $\neq$ and number of columns.
So a matrix $A=\left[a_{i j}\right] m \times n$ is said to be a rectangular matrix when $\mathrm{m} \neq \mathrm{n}$.
$
\text { For example, }\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. It is represented by 'o'
$\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] \mathrm{m} \times \mathrm{n}$, where $\mathrm{a}_{\mathrm{ij}}=0$
For example, $\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right],\left[\begin{array}{llll}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right]$

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 $\mathrm{m}=\mathrm{n}$.

Diagonal matrix: $\left[\begin{array}{ccc}a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33}\end{array}\right]$
A diagonal matrix of order $\mathrm{n} \times \mathrm{n}$ having diagonal elements as $\mathrm{d}_1, \mathrm{~d}_2, \mathrm{~d}_3 \ldots \ldots \ldots, \mathrm{d}_{\mathrm{n}}$ is denoted by $\operatorname{diag}\left[d_1, d_2, d_3 \ldots \ldots . ., d_n\right]$
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.
$
A=\left[\begin{array}{ll}
3 & 0 \\
0 & 3
\end{array}\right] \quad 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 $\mathrm{c}$ is not equal to 0

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

So, a square matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n}}$ is an 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]
$

The product of the conjugate transpose of a unitary matrix, with the unitary matrix, gives an identity matrix.

Upper triangular matrix: A square matrix whose all elements below the principal diagonal are zero is called an upper triangular matrix.
or $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be upper triangular if $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}=0$ when $\mathrm{i}>\mathrm{j}$.
$
\text { Upper triangular matrix: }\left[\begin{array}{ccccc}
a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
0 & a_{22} & a_{23} & a_{24} & a_{25} \\
0 & 0 & a_{33} & a_{34} & a_{35} \\
0 & 0 & 0 & a_{44} & a_{45} \\
0 & 0 & 0 & 0 & a_{55}
\end{array}\right]
$

Lower triangular matrix: A square matrix whose all elements above the principal diagonal is zero is called a lower triangular matrix.

Lower triangular matrix:
$
\left[\begin{array}{ccccc}
a_{11} & 0 & 0 & 0 & 0 \\
a_{21} & a_{22} & 0 & 0 & 0 \\
a_{31} & a_{32} & a_{33} & 0 & 0 \\
a_{41} & a_{42} & a_{43} & a_{44} & 0 \\
a_{51} & a_{52} & a_{53} & a_{54} & a_{55}
\end{array}\right]
$

Symmetric matrix: A square matrix $A=\left[a_{i j}\right]_{n \times n}$ is said to be symmetric if $A^{\prime}=A$,
$
\begin{aligned}
& \text { i.e., } \mathrm{a}_{\mathrm{ij}}=\mathrm{a}_{\mathrm{ji}} \forall \mathrm{i}, \mathrm{j} \\
& \mathrm{A}=\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right] \text { then } \mathrm{A}^{\prime}=\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right]
\end{aligned}
$

Clearly, $A=A^{\prime}$, hence $A$ is a symmetric matrix

Skew-symmetric matrix:
A square matrix $A=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be skew-symmetric if $\mathrm{A}^{\prime}=-\mathrm{A}$
$
\text { i.e. } A^{\prime}=-A \text {, i.e., } a_{i j}=-a_{j i} \forall i, j
$

Now if we put $\mathrm{i}=\mathrm{j}$, we have
$
\begin{aligned}
& \mathrm{a}_{\mathrm{ii}}=-\mathrm{a}_{\mathrm{ii}}, \\
& \therefore 2 \mathrm{a}_{\mathrm{ii}}=0 \Rightarrow \mathrm{a}_{\mathrm{ii}}=0 \forall \mathrm{i}^{\prime} \mathrm{s}
\end{aligned}
$

Hermitian matrix
A square matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] \mathrm{n} \times \mathrm{n}$ is said to be a Hermitian matrix if $a_{i j}=\overline{a_{j i}} \forall \mathrm{i}, \mathrm{j}$,
i.e. $A=A^\theta, \quad\left[\right.$ where $A^\theta$ is conjugate transpose of matrix $\left.A\right]$

Skew-hermitian matrix
A square matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right] \mathrm{n} \times \mathrm{n}$ is said to be a Skew-Hermitian matrix if $a_{i j}=-\overline{a_{i j}} \forall \mathrm{i}, \mathrm{j}$, i.e. $\mathrm{A}^\theta=-\mathrm{A}, \quad\left[\right.$ where $\mathrm{A}^\theta$ is conjugate transpose of matrix $\left.\mathrm{A}\right]$

Orthogonal matrix
A square matrix is said to be an orthogonal matrix if $A A^{\prime}=I$, where $I$ is the identity matrix.

Idempotent matrix
A square matrix is said to be an idempotent matrix if it satisfies the condition $A^2=A$

Nilpotent matrix

If A satisfies the condition Ak = O and Ak-1 ≠ O, then A is called a nilpotent matrix k is known as the order of nilpotent matrix A.

Involutory matrix
If A satisfies the condition $A^2=A$, where $\mathrm{I}$ is the identity matrix then $\mathrm{A}$ is called the involutory matrix.

Note: $A=A^{-1}$ for involutory matrix.

Periodic matrix

If a square matrix A satisfies the relation A^k+1 = A, where k is a +ve integer. Then A is called a periodic matrix. If k is the least +ve integer for which this condition is satisfied then k is called the period of A.

For k = 1, we get A² = A, which is the condition for the idempotent matrix, so the period of the idempotent matrix =1.

Summary

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Solved Examples Based On Types Of Matrices

Example 1: If m x n (where m and n are natural numbers) shows the order of a rectangular matrix, then the relation between m and n can be

1) m>n

2) m<n

3)

4) All of the above

Solution:

Rectangular matrix is the matrix in which is the number of rows $\neq$ number of columns.
So a matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be a rectangular matrix when $\mathrm{m} \neq \mathrm{n}$.

So the value of m and n can be any natural number but they are not equal to each other.

Hence, the answer is option 3.

Example 2: If matrix $A=\operatorname{diag}\left[\begin{array}{llll}3 & 5 & 7 & 8\end{array}\right]$ and matrix $B=\left[\begin{array}{cccc}5 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2\end{array}\right]$. Then $A-B=$

Solution:

Diagonal Matrix: A square matrix all of whose non-diagonal elements are Zero
$
\begin{aligned}
& \text { - wherein } \\
& A=\left[a_{i j}\right]_{n \times n} \\
& a_{i j}=0
\end{aligned}
$

Whenever $i \neq j$
$
\mathbf{D}=\left(\begin{array}{cccc}
\lambda_1 & 0 & 0 & \cdots \\
0 & \lambda_2 & 0 & \cdots \\
0 & 0 & \lambda_3 & \cdots \\
\vdots & \vdots & \vdots & \ddots
\end{array}\right)
$
$A=\operatorname{diag}\left[\begin{array}{llll}3 & 5 & 7 & 8\end{array}\right]$ and
$
B=\operatorname{diag}\left[\begin{array}{llll}
5 & 3 & 1 & -2
\end{array}\right]
$

Then $A-B=\operatorname{diag}\left[\begin{array}{llll}-2 & 2 & 6 & 10\end{array}\right]$
we just subtract corresponding elements
Hence $A-B=\operatorname{diag}\left[\begin{array}{llll}-2 & 2 & 6 & 10\end{array}\right]$

Example 3: If $\mathrm{A}$ is a strictly triangular matrix of order $3 \times 3$ and $B=\operatorname{diag}\left[\begin{array}{lll}3 & 5 & 2\end{array}\right]$; Then $|\mathrm{AB}|=$
Solution:
Strictly triangular matrix -
$
a_{i i}=0 \text { for } 1 \leq i \leq n
$

Where $A=\left[a_{i j}\right]_{n \times n}$
Since diagonal elements of $A$ are 0 and $B$ is a diagonal matrix, If we multiply them we get a matrix with determinant 0 since the first column and last row have all elements $=0$

Hence, $|A B|=0$

Example 4: If $\mathrm{A}$ and $\mathrm{B}$ are both square matrices of order $2 \times 2$ and $\mathrm{C}$ is a matrix with 5 columns then what should be the order of $\mathrm{D}$ such that $C D=A$

Solution:
In the square matrix,
Number of rows $=$ Number of columns
- wherein

A $m \times n$ matrix Where $m=n$
$C_{a * 5}$...and.. $B$ is of order 2*2
Thus $C_{a \times 5} \times D_{5 \times 2}=A_{2 \times 2}$

Thus a = 2 ;

and D has order 5 x 2 in order to get a square matrix

Hence the order of D is 5 x 2

Example 5: If A is an identity matrix of order 3 and B is a diagonal matrix of order 3 x 3. Then which of the following is correct?

1) $a_{11}=b_{12}$
2) $a_{13}=b_{23}$
3) $a_{22}=b_{22}$
4) $a_{12}=b_{22}$

Solution:
For identity matrix $a_{i j}=0 \forall i \neq j$ and $a_{i j}=1$
For diagonal matrix $a_{i j}=0 \forall i \neq j$
Thus $a_{13}=b_{23}=0$
Hence, the answer is option 2