| Operation | Rule | Caution |
|---|---|---|
| A + B | Commutative: A+B = B+A | Same order required |
| AB | NOT commutative: AB != BA generally | Same inner dimensions |
| A(BC) | Associative: A(BC) = (AB)C | Always valid |
| A(B+C) | Distributive: A(B+C) = AB + AC | Left and right distributive |
| (AB)^T | = * | Order reverses |
| (AB)^(-1) | = B^(-1) * A^(-1) | Order reverses |
| ()^(-1) | = (A^(-1))^n | Both exist |
| AB = 0 | Does NOT imply A=0 or B=0 | Unlike scalar algebra |
| = A | Does NOT imply A=I or A=0 | Idempotent matrices exist |
| AB = AC | Does NOT imply B=C | Cancellation needs A^(-1) |
The "reversal law" for transpose and inverse is tested every year in some form.