# Null Matrix

If each element of an m × n matrix be 0, the null element of F, the matrix is said to be the null matrix or the zero matrix of order m × n and it is denoted by Om,n. It is also denoted by O, when no confusion regarding its order arises.

Null or zero Matrix: Whether A is a rectangular or square matrix, A - A is a matrix whose every element is zero. The matrix whose every element is zero is called a null or zero matrix and it is denoted by 0.

Thus for A and 0 of the same order we have A + 0 = A

For example,

$$\begin{bmatrix} 0 & 0 \end{bmatrix}$$ is a zero matrix of order 1 × 2.

$$\begin{bmatrix} 0\\ 0 \end{bmatrix}$$ is a zero or null matrix of order 2 × 1.

$$\begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}$$ is a null matrix of order 2 × 2.

$$\begin{bmatrix} 5 & 6 & 4\\ 1 & 0 & 9 \end{bmatrix}$$ is a null matrix of order 2 × 3.

Problems on Null or zero matrix:

1. Find two nonzero matrices whose product is a zero matrix.

Solution:

Let A = $$\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$ and B = $$\begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}$$ be two non-zero matrices.

But AB = $$\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$ $$\begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}$$ = $$\begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}$$ is a null matrix.

2. If A = $$\begin{bmatrix} 1 & 2\\ -1 & -1 \end{bmatrix}$$, show that A2 + I = 0.

(I and 0 being identity and null matrices of order 2).

Solution:

Given, A = $$\begin{bmatrix} 1 & 2\\ -1 & -1 \end{bmatrix}$$

Now A2 = $$\begin{bmatrix} 1 & 2\\ -1 & -1 \end{bmatrix}$$$$\begin{bmatrix} 1 & 2\\ -1 & -1 \end{bmatrix}$$ = $$\begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix}$$

Therefore, A2 + I = $$\begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix}$$ + $$\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$$ = $$\begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}$$

Thus, A2 + I = 0.