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.
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