Manipal MET2022MathematicsMatrices
If A and n n non-singular matrix such that A A^T=A⁻¹ A and B=A⁻¹ A^T , then B B^ is equal to
Options
- A1+B
- B1
- CB⁻¹
- D(B⁻¹ )^
Correct answer
B. 1
Step-by-step solution
If A is non-singular matrix, then |A| 0 aligned A A^T & =A⁻¹ A and B=A⁻¹ A^T B B^ & = (A⁻¹ A^T ) (A⁻¹ A^T )^T & =A⁻¹ A^T A (A⁻¹ )^T [ (A I)^ =A ] & =A⁻¹ A A^T (A⁻¹ )^T & =I A^T (A⁻¹ )^T & =A^T (A⁻¹ )^T & = (A⁻¹ A )^T [ (A B)^T=B^T A^T ] & =I^T=1 aligned