Identify the correct statements.
(A) If A is a non-sigular matrix, then $A^{-1}=\frac{|A|}{(adjA)}$
(B) If A is an invertible matrix then $\frac{1}{|A^{-1}|}=|A|$
(C) If A and B are two invertible matrices of the same order then AB is also invertible matrix and $(BA)^{-1}=A^{-1}B^{-1}$
(D) If A is an invertible matrix, then AT is also invertible and $(A^T)^{-1}=\frac{1}{(A^{-1})^T}$
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 1
Option B is correct as $|A^n=|A|^n$ n be any integer.
$⇒\frac{1}{|A^{-1}|}=|A|$
Option C is also correct as if x, y are two matrices.
$xy = yx = 1$, then $x = y^{-1}$
Given $(BA)^{-1}=A^{-1}B^{-1}$
Multiply with BA to $A^{-1}B^{-1}$
$BAA^{-1}B^{-1}=B(AA^{-1})B^{-1}=BB^{-1}=I$
Multiply with $A^{-1}B^{-1}$ with BA
$A^{-1}B^{-1}BA=A^{-1}IA=A^{-1}A+I$
$BA(A^{-1}B^{-1})=(A^{-1}B^{-1})BA=I$
$(BA)^{-1}=A^{-1}B^{-1}$
So, statement B and C are correct.