What is Square Matrix? Examples & Properties

A square matrix is a type of matrix in which number of rows is equal to number of columns. Matrix P = [xij]m x n is said to be a square matrix when m = n. Here m is the number of rows and n is the number of columns. A square matrix P of order n has n rows and n columns.

The elements xij (such that i = j) i.e. x11, x22, x33,………, xnn are called diagonal elements. The elements xij (such that i ≠ j) i.e. x12, x21, x13,……… are called off-diagonal elements. The line along which diagonal elements lie is known as diagonal of matrix or principle diagonal of matrix.

Square Matrix Examples

The example of a 3 x 3 square matrix is given below.

    \[Q=\begin{bmatrix} 2 &1 &3 \\ 4& 5 &6 \\ 7&9 & 8 \end{bmatrix}\]

The above matrix Q represents a square matrix. The diagonal elements are 2, 5, and 8. The off-diagonal elements are 1, 3, 4, 6, 7, and 9.

The example of a 4 x 4 square matrix is given below.

    \[R=\begin{bmatrix} 7 &9 &6 &8 \\ 4& 5 &3 &2 \\ 0&1 & 10 &11 \\ 12&13 & 14 &15 \end{bmatrix}\]

The above matrix R represents a square matrix. The diagonal elements are 7, 5, 10 and 15. The off-diagonal elements are 9, 6, 8, 4, 3, 2, 0, 1, 11, 12, 13 and 14.

The example of a 2 x 2 square matrix is given below.

    \[C=\begin{bmatrix} 3 &4 \\ 8& 7 \end{bmatrix}\]

The above matrix C represents a square matrix. The diagonal elements are 3 and 7. The off-diagonal elements are 4 and 8.

Square Matrix Properties

1. In this matrix number of rows is equal to number of columns.

2. The determinant of a matrix can only be calculated for a square matrix.

3. Trace of a matrix is equal to the sum of diagonal elements of the square matrix.

4. Inverse of matrix is calculated only for a square matrix.

Square matrix types

The special kinds of a square matrix are

  1. Diagonal matrix
  2. Idempotent matrix
  3. Involutory matrix
  4. Lower triangular matrix
  5. Nilpotent matrix
  6. Non-singular matrix
  7. Scalar matrix
  8. Singular matrix
  9. Unit matrix or Identity matrix
  10. Upper triangular matrix

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