If A and B be two square matrices, then:(a) AB = BA(b) (AB)−1 = A−1 B−1(c) (AB)T = BTAT(d) none of these
Question
If A and B be two square matrices, then:(a) AB = BA(b) (AB)−1 = A−1 B−1(c) (AB)T = BTAT(d) none of these
Solution
(a) AB = BA: This statement is not always true. In general, matrix multiplication is not commutative, meaning that the order in which matrices are multiplied can affect the result. So, AB is not necessarily equal to BA.
(b) (AB)−1 = A−1 B−1: This statement is also not true. The correct relation is (AB)−1 = B−1 A−1. This is known as the reverse order law for inverses of matrices.
(c) (AB)T = BTAT: This statement is true. This is a property of the transpose of a product of matrices. The transpose of a product of two matrices is equal to the product of their transposes in the reverse order.
(d) none of these: Since statement (c) is true, this option is not correct.
So, the correct answer is (c) (AB)T = BTAT.
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