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,
[00] is a zero matrix of order 1 × 2.
[00] is a zero or null matrix of order 2 × 1.
[0000] is a null matrix of order 2 × 2.
[564109] 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 = [1001] and B = [0001] be two non-zero matrices.
But AB = [1001] [0001] = [0000] is a null matrix.
2. If A = [12−1−1], show that A2 + I = 0.
(I and 0 being identity and null matrices of order 2).
Solution:
Given, A = [12−1−1]
Now A2 = [12−1−1][12−1−1] = [−100−1]
Therefore, A2 + I = [−100−1] + [1001] = [0000]
Thus, A2 + I = 0.
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