Square Matrix

If square matrixes have n rows or columns then the matrix is called the square matrix of order n or an n-square matrix.

Definition of Square Matrix: An n × n matrix is said to be a square matrix of order n. In other words when the number of rows and the number of columns in the matrix are equal then the matrix is called square matrix.

For example:

$A = \begin{bmatrix} 2 & 7 & 9\\ 1 & 4 & 2\\ 8 & 6 & 3 \end{bmatrix}$

The number of rows of the above matrix = 3

The number of columns of the above matrix = 3

Since the number of rows and the number of columns are equal, the above matrix A is a square matrix.

Again, let us consider an another matrix $Z = \begin{bmatrix} 2 & 5 & 6\\ 1 & 7 & 9 \end{bmatrix}$

The number of rows of the above matrix = 2

The number of columns of the above matrix = 3

Since the number of rows and the number of columns are not equal, the above matrix Z is not a square matrix.

The diagonal through the left hand top cornet element of the square matrix is said to be the principal diagonal of the matrix and the elements in the principal diagonal are said to be the diagonal elements of the square matrix.

Let us consider some examples:

1. $A = \begin{bmatrix} {\color{Red} 8} & 5\\ 3 & {\color{Red} 6} \end{bmatrix}$

In the above 2 ×2 square matrix A the diagonal 8, 6 is the principal diagonal and 8 and 6 are said to be the diagonal elements.

2. $Z = \begin{bmatrix} {\color{Red} 7} & 3 & 2\\ 5 & {\color{Red} 2} & 3\\ 1 & 0 & {\color{Red} 6} \end{bmatrix}$

In the above 3 × 3 square matrix Z the diagonal 7, 2, 6 is the principal diagonal and 7, 2 and 6 are said to be the diagonal elements.

3. $P = \begin{bmatrix} {\color{Red} 8} & 5 & 7 & 1\\ 2 & {\color{Red} 1} & 6 & 8\\ 7 & 3 & {\color{Red} 0} & 5\\ 2 & 8 & 6 & {\color{Red} 4} \end{bmatrix}$

In the above 4 × 4 square matrix P the diagonal 8, 1, 0, 4 is the principal diagonal and 8, 1, 0 and 4 are said to be the diagonal elements.