We need to form matrices A, B and C such that AB = AC, where A is a non-zero matrix, but B ≠ C.
Take,
Multiply 1st row of matrix A by matching members of 1st column of matrix B, then sum them up.
(1, 0)(1, 2) = (1 × 1) + (0 × 2)
⇒ (1, 0)(1, 2) = 1 + 0
⇒ (1, 0)(1, 2) = 1
Multiply 1st row of matrix A by matching members of 1st column of matrix C, then sum them up.
(1, 0)(1, 2) = (1 × 1) + (0 × 2)
⇒ (1, 0)(1, 2) = 1 + 0
⇒ (1, 0)(1, 2) = 1
Clearly, AB = AC.
Thus, we have found an example that satisfy the required criteria.