Q. 165.0( 2 Votes )
Show by an example that for A ≠ O, B ≠ O, AB = O.
We know that,
In order to multiply two matrices, A and B, the number of columns in A must equal the number of rows in B. Thus, if A is an m x n matrix and B is an r x s matrix, n = r.
We are given that,
A ≠ 0 and B ≠ 0
We need to show that, AB = 0.
For multiplication of A and B,
Number of columns of matrix A = Number of rows of matrix B = 2 (let)
Matrices A and B are square matrices of order 2 × 2.
For AB to become 0, one of the column of matrix A and other row of matrix B must be 0.
Check: Multiply AB.
Multiply 1st row of matrix A by matching members of 1st column of matrix B, then sum them up.
(0, 1).(3, 0) = (0 × 3) + (1 × 0)
⇒ (0, 1).(3, 0) = 0 + 0 = 0
Similarly, let us do it for the rest of the elements.
Thus, this example justifies the criteria.
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