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.

For example,

Check: Multiply AB.

Multiply 1^st row of matrix A by matching members of 1^st 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.