What is Zero Matrix or Null Matrix? Examples & Properties

Zero Matrix is a type of matrix whose elements are equal to zero. Zero matrix is also known as null matrix. It is generally denoted by capital letter ‘O‘.

A matrix O = [xij] is said to be null matrix or zero matrix if xij = 0 for all values of ‘i‘ and ‘j‘. This matrix need not to be a square matrix.

Null matrix examples

The example of a null matrix of order 3 (or matrix size is 3 x 3) is given below.

    \[O=\begin{bmatrix} 0 &0 &0 \\ 0& 0 &0 \\ 0&0 & 0 \end{bmatrix}\]

The example of a zero matrix of order 2 (or matrix size is 2 x 2) is given below.

    \[O=\begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix}\]

The example of a null matrix of size 3 x 1 is given below.

    \[O=\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}\]

The example of a zero matrix of size 2 x 3 is given below.

    \[\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\]

Null matrix properties

Let P be a matrix and O is a null matrix of the same size as of matrix P.

1. P + O = O + P = P

2. P + (-P) = O

3. The determinant of a null matrix is 0 i.e. |O| = 0.

4. The rank of a null matrix or zero matrix is zero.

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