Let A and B be real matrices of the form [ array ll & 0 0 & array ] and [ array ll 0 & & 0 array ] , respectively. Statement 1: A B-B A is always an invertible matrix. Statement 2: A B-B A is never an identity matrix.

Step-by-step solution

Let A and B be real matrices such that A= [ array ll & 0 0 & array ] and B= [ array ll 0 & & 0 array ] Now, A B= [ array cc 0 & & 0 array ] and B A= [ array cc 0 & & 0 array ] Statement-1: aligned & .A B-B A= [ array cc 0 & ( - ( - ) & 0 array ] ) & |A B-B A|= ( - ^2 0 aligned A B-B A is always an invertible matrix. Hence, statement -1 is true. But A B-B A can be identity matrix if =- or =- So, statement -2 is false.