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 where is the rows and is number of columns in the matrix.

If is a matrix of order , 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 where is known as **square matrix.** For example,

A = \begin{bmatrix}1 & 4 & 9\\ 3 & 7 & 3 \\ 1 & 3 & 1\end{bmatrix}_{3 \times 3}

Note: 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}

**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**.

\begin{aligned} &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} \end{aligned}

**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.

\begin{aligned} &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} \end{aligned}

**Null Matrix**

Usually no one creates null matrix, it is obtained due to some algebraic operations performed in matrices. A matrix with all elements equal to zero is called **a null matrix**.

\begin{aligned} &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} \end{aligned}

**Upper Triangular Matrix**

Matrix with lower diagonals equal to zero is called **an Upper triangular matrix.** For example,

\begin{aligned} &U_{3 \times 3} = \begin{bmatrix}1 & 2 & 3\\ 0 & 4 & 5 \\ 0 & 0 & 6\end{bmatrix}_{3 \times 3} \end{aligned}

For upper triangular matrix, , for and denoted by .

**Lower Triangular Matrix**

The lower triangular matrix has its upper diagonals as zero and it is denoted by .

\begin{aligned} L_{3 \times 3} = \begin{bmatrix}1 & 0 & 0\\ 4 & 2 & 0 \\ 7 & 3 & 3\end{bmatrix}_{3 \times 3} \end{aligned}

For lower triangular matrix, , for and denoted by .

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