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 O_{m,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 A^{2} + 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 A^{2 }= \(\begin{bmatrix} 1 & 2\\ -1 & -1
\end{bmatrix}\)\(\begin{bmatrix} 1 & 2\\ -1 & -1 \end{bmatrix}\) = \(\begin{bmatrix}
-1 & 0\\ 0 & -1 \end{bmatrix}\)

Therefore, A^{2 }+ 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, A^{2} + I = 0.

**Didn't find what you were looking for? Or want to know more information
about Math Only Math.
Use this Google Search to find what you need.**

## New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.