Show by an example of 2 × 2 matrices that AB = O does not imply A = O or B = O, and that AB = O does not imply BA = O.

1. 1 · Write what is given and what is to be found. To show: there are non-zero A and B with AB = O, and for the same pair BA ≠ O.
2. 2 · Choose the matrices. Take A = [1, 2; 2, 4] and B = [2, −6; −1, 3]. Neither is the zero matrix.
3. 3 · Write the rule. Multiply row by column: the (i, k) entry is row i of the first matrix times column k of the second.
4. 4 · Find AB.
   • Row 1: (1)(2) + (2)(−1) = 0 and (1)(−6) + (2)(3) = 0
   • Row 2: (2)(2) + (4)(−1) = 0 and (2)(−6) + (4)(3) = 0
   • AB = O
5. 5 · Find BA.
   • Row 1: (2)(1) + (−6)(2) = −10 and (2)(2) + (−6)(4) = −20
   • Row 2: (−1)(1) + (3)(2) = 5 and (−1)(2) + (3)(4) = 10
   • BA = [−10, −20; 5, 10] ≠ O
6. 6 · Check and box the answer. A ≠ O and B ≠ O, yet AB = O; and BA ≠ O. So AB = O implies neither 'A = O or B = O' nor 'BA = O'.

Scoring tip: State clearly that A ≠ O and B ≠ O before you multiply. Without that line the example proves nothing.