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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,

[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 = [1211], show that A2 + I = 0.

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

Solution:

Given, A = [1211]

Now A2 = [1211][1211] = [1001]

Therefore, A2 + I = [1001] + [1001] = [0000]

Thus, A2 + I = 0.





10th Grade Math

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