Consider the following in respect of non-singular matrices A and B : I. (AB)⁻¹ = A⁻¹B⁻¹ II. (BA)(AB)⁻¹ = I , where I is the identity matrix III. (AB)^T = A^T B^T How many of the above are correct?

Step-by-step solution

For Statement I, the reversal law of matrix inverse gives (AB)⁻¹ = B⁻¹A⁻¹ . Since matrix multiplication is not commutative in general, B⁻¹A⁻¹ A⁻¹B⁻¹ . Thus, Statement I is incorrect. For Statement II, using the reversal law, (BA)(AB)⁻¹ = (BA)(B⁻¹A⁻¹) = B(AB⁻¹)A⁻¹ = BIA⁻¹ = BA⁻¹ . This is not equal to I in general. Thus, Statement II is incorrect. For Statement III, the reversal law of matrix transpose gives (AB)^T = B^T A^T . Since matrix multiplication is not commutative, B^T A^T A^T B^T in general. Thus, Statemen