Types of Matrices

Matrices

A matrix is a rectangular arrangement of numbers or elements in rows and columns. It is a useful tool in mathematics to organize and solve problems related to trace of matrix, order, etc. in areas like algebra, geometry, and even computer science.

Below is an example of a matrix structure of 3 rows and 4 columns:

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

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

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

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

Types of Matrices

Matrices are categorized into different types based on their structure and the way their elements are arranged. There are several types of matrices, which show how the matrix behaves in different matrix operations. We have provided detailed explanations about each type of matrix along with their curated examples, to help you understand better.

Row Matrix

A matrix is known as a row matrix when it has just one row. In simple words, we can say that al the elements are arranged in a row.

For a matrix $A = [a_{ij}]_{m \times n}$ to be a row matrix, the number of rows must be 1 i.e., $m = 1$.

Denoted by: $A = \left[\begin{array}{lllll} a_{11} & a_{12} & a_{13} & \ldots & a_{1n} \end{array}\right]_{1 \times n}$

Example:

$\left[\begin{array}{llll} 1 & 32 & 81 & -32 \end{array}\right]$

This is a row matrix with 1 row and 4 columns — so, its order is $1 \times 4$.

Suppose a student scores 76, 85, 93, and 88 in four subjects. These values can be arranged in a row matrix like this:

$M = \left[\begin{array}{llll} 76 & 85 & 93 & 88 \end{array}\right]_{1 \times 4}$

This makes it easier to view the scores as a single set of values across multiple categories.

Column Matrix

A matrix which has all its elements arranged in a single column is known as a column matrix. Similar to the row matrix, the column matrix has all its elements arranged vertically.

For the matrix $B = [a_{ij}]_{m \times n}$, to be a column matrix, the number of columns must be 1 i.e. $n = 1$.

Denoted by: $A = \left[\begin{array}{c} a_{11} \\ a_{21} \\ a_{31} \\ \vdots \\ a_{m1} \end{array}\right]_{m \times 1}$

This shows that there is only one column and m rows.

Example:

$\left[\begin{array}{r} 7 \\ -3 \\ 10 \end{array}\right]$

This is a column matrix with 3 rows and 1 column — so, its order is $3 \times 1$.

Suppose the temperature on three consecutive days was 30°C, 32°C, and 31°C. These values can be written as a column matrix:

$T = \left[\begin{array}{r} 30 \\ 32 \\ 31 \end{array}\right]_{3 \times 1}$

This makes it easy to organize and analyze the data over multiple time periods.

Note: A matrix with just one row or one column is known as a vector. When the matrix has a single row, it's called a row vector, and when it has a single column, it's called a column vector.

Square Matrix

The matrix which has the same number of rows and same number of columns is called as the square matrix.

For a matrix $A = [a_{ij}]_{m \times n}$, to be a square matrix, it should satisfy the condition $m = n$.

Denoted by: $A = \left[\begin{array}{ccc} a_{11} & a_{12} & \cdots \\ a_{21} & a_{22} & \cdots \\ \vdots & \vdots & \ddots \end{array}\right]_{n \times n}$

This means the matrix has n rows and n columns — the same number on both sides.

Example:

$\left[\begin{array}{rr} 5 & -2 \\ 7 & 4 \end{array}\right]$

This is a square matrix of order $2 \times 2$, as it has 2 rows and 2 columns.

Imagine you have a chart that shows the number of boys and girls in two different sections of a class. You could represent that information using a square matrix like this:

$S = \left[\begin{array}{rr} 12 & 15 \\ 10 & 13 \end{array}\right]_{2 \times 2}$

Each row could represent a section, and each column could represent boys and girls.

Note: Square matrices also allow special types like:

• Symmetric matrix: where $A = A'$ (i.e., matrix equals its transpose)

• Skew-symmetric matrix: where $A = -A'$

Any square matrix $A$ can be written as a combination of both:

$A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')$

Here, the first part is symmetric, and the second part is skew-symmetric.

The adjoint and inverse of a square matrix can be calculated.

Rectangular Matrix

A matrix in which the number of rows is not equal to the number of columns is called a rectangular matrix. That is, for a matrix $A = [a_{ij}]_{m \times n}$, if $m \ne n$, it is a rectangular matrix.

Denoted by: $A = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{array}\right]_{2 \times 3}$ (example form)

Example:

$\left[\begin{array}{rrr} 2 & 4 & 6 \\ 1 & 3 & 5 \end{array}\right]$
This is a rectangular matrix of order $2 \times 3$.

A student records scores of 2 students across 3 tests. Represent this data as a matrix.
$R = \left[\begin{array}{rrr} 78 & 85 & 90 \\ 66 & 73 & 80 \end{array}\right]_{2 \times 3}$

Null (Zero) Matrix

A matrix in which all the elements are zero is called a null matrix or zero matrix. A matrix $A = [a_{ij}]_{m \times n}$ is a zero matrix if $a_{ij} = 0$ for all $i$ and $j$.

Denoted by: $A = \left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]_{2 \times 3}$ (example form)

Example:

$\left[\begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array}\right]$
This is a $2 \times 2$ zero matrix.

A machine failed to produce any units over two shifts. Represent this using a matrix.
$Z = \left[\begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array}\right]_{2 \times 2}$

Diagonal Matrix

A square matrix in which all the non-diagonal elements are zero is called a diagonal matrix.
A matrix $A = [a_{ij}]_{m \times n}$ is diagonal if $a_{ij} = 0$ for all $i \ne j$.

Denoted by: $A = \left[\begin{array}{ccc} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{array}\right]$

Example:

$\left[\begin{array}{ccc} 3 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & 8 \end{array}\right]$
This is a $3 \times 3$ diagonal matrix.

Represent the weights (in kg) of three packages placed independently on a scale.
$W = \left[\begin{array}{ccc} 10 & 0 & 0 \\ 0 & 15 & 0 \\ 0 & 0 & 12 \end{array}\right]$

Symmetric Matrix

A square matrix is called symmetric if it is equal to its transpose, i.e., $A = A^T$.
If $A = [a_{ij}]_{m \times n}$ and $a_{ij} = a_{ji}$ for all $i$ and $j$, then $A$ is symmetric.

Denoted by: $A = \left[\begin{array}{ccc} a & b & c \\ b & d & e \\ c & e & f \end{array}\right]$
Here, $a_{ij} = a_{ji}$

Example:

$\left[\begin{array}{ccc} 4 & 2 & -1 \\ 2 & 5 & 3 \\ -1 & 3 & 6 \end{array}\right]$
This is a symmetric matrix.

A correlation matrix where relationships between variables are mutual is symmetric in nature.

Skew-Symmetric Matrix

A square matrix is called skew-symmetric if its transpose is equal to the negative of the matrix, i.e., $A^T = -A$.

If $A = [a_{ij}]_{m \times n}$ and $a_{ij} = -a_{ji}$ for all $i$ and $j$, and $a_{ii} = 0$, then $A$ is skew-symmetric.

Denoted by: $A = \left[\begin{array}{ccc} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{array}\right]$

Example:

$\left[\begin{array}{ccc} 0 & 3 & -5 \\ -3 & 0 & 2 \\ 5 & -2 & 0 \end{array}\right]$
This is a skew-symmetric matrix.

In physics, such matrices appear in cross product and rotation-related calculations in vectors.

Scalar Matrix

A diagonal matrix in which all the diagonal elements are equal is called a scalar matrix.

A square matrix $A = [a_{ij}]_{m \times n}$ is scalar if $a_{ij} = 0$ for $i \ne j$ and $a_{ii} = k$ (same constant) for all $i$.

Denoted by: $A = \left[\begin{array}{ccc} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{array}\right]$

Example:

$\left[\begin{array}{ccc} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right]$
This is a scalar matrix of order $3 \times 3$.

A company assigns the same budget to three departments. Represent this allocation using a scalar matrix:
$B = \left[\begin{array}{ccc} 10 & 0 & 0 \\ 0 & 10 & 0 \\ 0 & 0 & 10 \end{array}\right]$

Identity (or Unitary) Matrix

A square matrix with 1’s on the main diagonal and 0’s elsewhere is called an identity matrix or unitary matrix.

A matrix $I = [a_{ij}]_{m \times n}$ where $a_{ij} = 1$ if $i = j$, and $a_{ij} = 0$ if $i \ne j$.

Denoted by: $I_n = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$

Example:

$\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$
This is the identity matrix of order $3 \times 3$.

In matrix multiplication, the identity matrix behaves like the number 1. For example, $A \cdot I = A$ for any square matrix $A$.

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

Upper Triangular Matrix

A square matrix in which all the elements below the main diagonal are zero is called an upper triangular matrix.
A matrix $A = [a_{ij}]_{m \times n}$ is upper triangular if $a_{ij} = 0$ for all $i > j$.

Denoted by: $A = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ 0 & a_{22} & a_{23} \\ 0 & 0 & a_{33} \end{array}\right]$

Example:

$\left[\begin{array}{ccc} 3 & 5 & -2 \\ 0 & 4 & 7 \\ 0 & 0 & 6 \end{array}\right]$
This is a $3 \times 3$ upper triangular matrix.

In solving linear equations, upper triangular matrices simplify back-substitution steps.

Lower Triangular Matrix

A square matrix in which all the elements above the main diagonal are zero is called a lower triangular matrix.
A matrix $A = [a_{ij}]_{m \times n}$ is lower triangular if $a_{ij} = 0$ for all $i < j$.

Denoted by: $A = \left[\begin{array}{ccc} a_{11} & 0 & 0 \\ a_{21} & a_{22} & 0 \\ a_{31} & a_{32} & a_{33} \end{array}\right]$

Example:

$\left[\begin{array}{ccc} 4 & 0 & 0 \\ -2 & 3 & 0 \\ 1 & 5 & 7 \end{array}\right]$
This is a $3 \times 3$ lower triangular matrix.

Used in forward substitution when solving linear equations involving matrix factorization.

Know more about: Lower & Upper triangular matrix

Hermitian Matrix

A Hermitian matrix is a square matrix whose conjugate transpose is equal to the original matrix, i.e., $A^H = A$.

A square matrix $A = [a_{ij}]_{m \times n}$ is Hermitian if $a_{ij} = \overline{a_{ji}}$ for all $i$ and $j$, where $\overline{a_{ji}}$ denotes the complex conjugate of $a_{ji}$.

Denoted by: $A^H = \overline{A^T} = A$

Example:

$\left[\begin{array}{cc} 3 & 2 + i \\ 2 - i & 4 \end{array}\right]$
This is a Hermitian matrix because the element at $(1,2)$ is the complex conjugate of the element at $(2,1)$, and the diagonal elements are real.

A Hermitian matrix often appears in quantum mechanics, where observable quantities (like energy, momentum) are represented by such matrices.

Skew-Hermitian Matrix

A Skew-Hermitian matrix is a square matrix whose conjugate transpose is the negative of the original matrix, i.e., $A^H = -A$.

A square matrix $A = [a_{ij}]_{m \times n}$ is skew-Hermitian if $a_{ij} = -\overline{a_{ji}}$ for all $i$ and $j$.

Denoted by: $A^H = \overline{A^T} = -A$

Example:

$\left[\begin{array}{cc} 0 & 3 + i \\ -3 + i & 0 \end{array}\right]$
This is a skew-Hermitian matrix because the element at $(2,1)$ is the negative complex conjugate of the element at $(1,2)$, and the diagonal elements are purely imaginary or zero.

In complex systems, skew-Hermitian matrices arise in differential equations and signal processing.

Know more about: Hermitian & Skew Hermitian Matrix

Orthogonal Matrix

A square matrix is said to be orthogonal matrix if its transpose is equal to its inverse, i.e., $A^T = A^{-1}$. A matrix $A$ is orthogonal if $A A^T = A^T A = I$, where $I$ is the identity matrix.

Denoted by: $A^T = A^{-1}$

Example:

$\left[\begin{array}{rr} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$
This is an orthogonal matrix for real values of $\theta$.

Idempotent Matrix

A matrix is called idempotent matrix if, when squared, it gives the same matrix, i.e., $A^2 = A$. A matrix $A$ is idempotent if $A^2 = A$

Denoted by: $A^2 = A$

Example:

$\left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right]$
This matrix satisfies $A^2 = A$, hence it is idempotent.

Nilpotent Matrix

A matrix is called nilpotent if some power of it results in the zero matrix, i.e., $A^k = 0$ for some positive integer $k$. A matrix $A$ is nilpotent if there exists a positive integer $k$ such that $A^k = 0$

Denoted by: $\exists\ k \in \mathbb{N}$ such that $A^k = 0$

Example:

$A = \left[\begin{array}{rr} 0 & 1 \\ 0 & 0 \end{array}\right]$
Here, $A^2 = \left[\begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array}\right] = 0$

Nilpotent matrices are used in solving systems of differential equations and matrix exponentials.

Involutory Matrix

A matrix is said to be involutory if its square is equal to the identity matrix, i.e., $A^2 = I$. A matrix $A$ is involutory if $A^2 = I$

Denoted by: $A^2 = I$

Example:

$A = \left[\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right]$
Here, $A^2 = I$, so $A$ is an involutory matrix.

Involutory matrices are useful in cryptography and coding theory for reversing transformations.

Equal Matrices

Two matrices are said to be equal if they have the same order and their corresponding elements are equal.
If $A = [a_{ij}]$ and $B = [b_{ij}]$ are matrices of order $m \times n$, then $A = B$ if and only if $a_{ij} = b_{ij}$ for all $i$ and $j$

Example:

Let $A = \left[\begin{array}{rr} 3 & 5 \\ 1 & -2 \end{array}\right]$,
$B = \left[\begin{array}{rr} 3 & 5 \\ 1 & -2 \end{array}\right]$
Here, $A = B$

If two identical survey tables are created by different people, and both have the same values in the same order, they are equal matrices.

Equivalent Matrices

Two matrices are said to be equivalent if they have the same order of matrix, but not necessarily the same elements.

If $A$ and $B$ are matrices of order $m \times n$, then they are equivalent if they have the same size: $A \sim B$ (symbolically).

Example:

$A = \left[\begin{array}{rr} 1 & 2 \\ 3 & 4 \end{array}\right]$,
$B = \left[\begin{array}{rr} 9 & 8 \\ 7 & 6 \end{array}\right]$
Here, $A \sim B$ (equivalent), but $A \ne B$.

Two matrices representing different datasets (like heights and weights) but with the same number of entries can be equivalent.

Singular Matrix

A square matrix is called a singular matrix if its determinant is equal to zero, i.e., $\det(A) = 0$. If $A$ is a square matrix and $\det(A) = 0$, then $A$ is singular.

Denoted by: $\det(A) = 0$

Example:

$A = \left[\begin{array}{rr} 2 & 4 \\ 1 & 2 \end{array}\right]$
Here, $\det(A) = (2)(2) - (4)(1) = 4 - 4 = 0$, so $A$ is singular.

Non-Singular Matrix

A matrix is called non-singular when its determinant is not equal to zero. In simple terms, it means the matrix is "invertible" — we can find its inverse.

Denoted by: $\det(A) \ne 0$

Example:

Let $A = \left[\begin{array}{rr} 3 & 5 \\ 2 & 7 \end{array}\right]$

Then,

$\det(A) = (3)(7) - (5)(2) = 21 - 10 = 11 \ne 0$, so $A$ is non-singular.

To solve a system of equations like $AX = B$, the matrix $A$ must be non-singular to find $X = A^{-1}B$.

Summary Table

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Solved Examples Based On Types Of Matrices

Question 1: If a matrix has an order of $m \times n$ (where m and n are natural numbers), what is the correct relationship between m and n for it to be a rectangular matrix?

1. $m > n$

2. $m < n$

3. $m \ne n$

4. All of the above

Solution: A rectangular matrix is one where the number of rows and columns are not equal. That means $m \ne n$. So, whether $m$ is more than $n$, or less than $n$, doesn't matter—as long as they're not equal.
Correct option: 3

Question 2: If $A = \operatorname{diag}\left[\begin{array}{llll}3 & 5 & 7 & 8\end{array}\right]$ and $B = \left[\begin{array}{cccc}5 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2\end{array}\right]$, then what is $A - B$?

Solution:
A diagonal matrix has values only on the diagonal (top-left to bottom-right), and everything else is zero.
In this case, matrix $B$ is just another way of writing
$B = \operatorname{diag}\left[\begin{array}{llll}5 & 3 & 1 & -2\end{array}\right]$.

Now to subtract $B$ from $A$, just subtract the diagonal elements one by one:
$[3 - 5,\ 5 - 3,\ 7 - 1,\ 8 - (-2)] = [-2,\ 2,\ 6,\ 10]$

So the result is:
$A - B = \operatorname{diag}\left[\begin{array}{llll}-2 & 2 & 6 & 10\end{array}\right]$

Answer: $\operatorname{diag}[-2,\ 2,\ 6,\ 10]$

Question 3: If $A$ is a strictly triangular matrix of order $3 \times 3$ and $B = \operatorname{diag}[3, 5, 2]$, then what is $|AB|$?

Solution: A strictly triangular matrix means that all diagonal elements are 0. So when we multiply it with a diagonal matrix, the resulting matrix still has zeros along at least one full row or column.

Since one row or column is completely zero, the determinant of the matrix becomes 0.

Answer: $|AB| = 0$

Question 5: If $A$ is an identity matrix of order 3 and $B$ is a diagonal matrix of order $3 \times 3$, 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:

• In an identity matrix, all diagonal elements are 1, and everything else is 0.

• In a diagonal matrix, diagonal elements can be any number, and off-diagonal elements are 0.

Let’s check each option:

• Option 1: $a_{11} = 1$, $b_{12} = 0$ i.e. Not equal

• Option 2: $a_{13} = 0$, $b_{23} = 0$ i.e. Both are 0 i.e. True

• Option 3: $a_{22} = 1$, $b_{22}$ could be anything i.e. Not always true

• Option 4: $a_{12} = 0$, $b_{22}$ could be non-zero i.e. Not always true

Correct option: 2