# Types Of Matrices

In the previous article, we learned about systems of linear equations can be represented using a matrix or augmented matrix. There are many types of matrices which we are going to explore in this post.

In general matrix are referred using their order which is $m \times n$ where $m$ is the rows and $n$ is number of columns in the matrix.

If $A[a_{ij}]$ is a matrix of order $m \times n$ , then, $A = [a_{ij}]_{m \times n}$ = $\begin{bmatrix} a_{11} & a_{12} & a_{1n}\\ a_{21} & a_{22} & a_{2n}\\ a_{m1} & a_{m2} & a_{mn}\end{bmatrix}$.

### Types of Matrices

There are many types of matrix in linear systems. We have listed few important ones.

• Square matrix
• Diagonal matrix
• Scalar matrix
• Unit or Identity matrix
• Null matrix
• Upper triangular
• Lower triangular matrix

There are some other types about which we shall discuss later. Let us try to know these basic matrices more.

Square Matrix

A matrix of order $m \times n$ where $m = n$ is known as square matrix. For example, $A = \begin{bmatrix}1 & 4 & 9\\ 3 & 7 & 3 \\ 1 & 3 & 1\end{bmatrix}_{3 \times 3}$ $A = \begin{bmatrix}1 & 4 & 9\\ 3 & 7 & 3 \\ 1 & 3 & 1\end{bmatrix}_{3 \times 3}$

Note: $a_{11}, a_{22}, a_{33}$ $a_{11}, a_{22}, a_{33}$ are diagonal elements.

Diagonal Matrix

A square matrix with all diagonal elements as 0 is called a diagonal matrix, but the diagonal elements may or may not be zero. For example, $A = \begin{bmatrix}1 & 0 & 0\\ 0 & 7 & 0 \\ 0 & 0 & 1\end{bmatrix}_{3 \times 3}$ $A = \begin{bmatrix}1 & 0 & 0\\ 0 & 7 & 0 \\ 0 & 0 & 1\end{bmatrix}_{3 \times 3}$

Scalar Matrix

A diagonal matrix with equal diagonal elements are called a scalar matrix. The scalar matrix is obtained by multiplying the identity matrix with a scalar value. $A = \begin{bmatrix}7 & 0 & 0\\ 0 & 7 & 0 \\ 0 & 0 & 7\end{bmatrix}_{3 \times 3}$ $A = \begin{bmatrix}7 & 0 & 0\\ 0 & 7 & 0 \\ 0 & 0 & 7\end{bmatrix}_{3 \times 3}$ $B= \begin{bmatrix}2 & 0\\ 0 & 2\end{bmatrix}_{2 \times 2}$ $B= \begin{bmatrix}2 & 0\\ 0 & 2\end{bmatrix}_{2 \times 2}$

Unit Matrix or Identity Matrix

A square matrix with diagonal elements as 1 and all non-diagonal elements as 0 is known as a Unit or an Identity matrix. Also, note that the unit matrix is a scalar matrix in itself. $I_3 = \begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}_{3 \times 3}$ $I_3 = \begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}_{3 \times 3}$ $I_2 = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}_{2 \times 2}$ $I_2 = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}_{2 \times 2}$

Null Matrix

Usually no one creates null matrix, it is obtained due to some algebraic operations performed in matrices. A $m \times n$ matrix with all elements equal to zero is called a null matrix. $O_3 = \begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}_{3 \times 3}$ $O_3 = \begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}_{3 \times 3}$ $O_2 = \begin{bmatrix}0 & 0\\ 0 & 0\end{bmatrix}_{2 \times 2}$ $O_2 = \begin{bmatrix}0 & 0\\ 0 & 0\end{bmatrix}_{2 \times 2}$

Upper Triangular Matrix

Matrix with lower diagonals equal to zero is called an Upper triangular matrix. For example, $U_{3 \times 3} = \begin{bmatrix}1 & 2 & 3\\ 0 & 4 & 5 \\ 0 & 0 & 6\end{bmatrix}_{3 \times 3}$ $U_{3 \times 3} = \begin{bmatrix}1 & 2 & 3\\ 0 & 4 & 5 \\ 0 & 0 & 6\end{bmatrix}_{3 \times 3}$

For upper triangular matrix, $a_{ij} = 0$, for $i > j$ and denoted by $U$.

Lower Triangular Matrix

The lower triangular matrix has its upper diagonals as zero and it is denoted by $L$. $L_{3 \times 3} = \begin{bmatrix}1 & 0 & 0\\ 4 & 2 & 0 \\ 7 & 3 & 3\end{bmatrix}_{3 \times 3}$ $L_{3 \times 3} = \begin{bmatrix}1 & 0 & 0\\ 4 & 2 & 0 \\ 7 & 3 & 3\end{bmatrix}_{3 \times 3}$

For lower triangular matrix, $a_{ij} = 0$, for $i < j$ and denoted by $L$.

In the next post, we shall discuss basic row operations on matrices which are very useful in solving system of linear equations.