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.
![\[O=\begin{bmatrix} 0 &0 &0 \\ 0& 0 &0 \\ 0&0 & 0 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-ce86bd946705857997dde7733726a0e7_l3.png "Rendered by QuickLaTeX.com")
The example of a zero matrix of order 2 (or matrix size is 2 x 2) is given below.
![\[O=\begin{bmatrix} 0 &0 \\ 0& 0 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-6d038761c66f987f978869c8ba1e91d8_l3.png "Rendered by QuickLaTeX.com")
The example of a null matrix of size 3 x 1 is given below.
![\[O=\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-8705c3b6015468d1ff0f1799e9be40b6_l3.png "Rendered by QuickLaTeX.com")
The example of a zero matrix of size 2 x 3 is given below.
![\[\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\]](https://cdn-0.electricalvoice.com/wp-content/ql-cache/quicklatex.com-ac07a23032a8b5d517c9ba7d61476c54_l3.png "Rendered by QuickLaTeX.com")
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.
