Question

If A is a $3\times 3$ non-singular matrix such that $A{A}^T=A^{T}AandB=A^{-1}{A}^T,thenB{B}^T$ is equal to

• A. I + B
• B. I
• C. $B^{-1}$
• D. ${\left(B^{-1}\right)}^T$

Answer 1

The correct option is B

I

Given, $A{A}^T=A^{T}AandB=A^{-1}{A}^T$
We are asked for the value of the expression $B{B}^T$. Since B is given in terms of A we will substitute for the values of $B^T$ and B.
$B{B}^T=\left(A^{-1}{A}^T\right)(A^{-1}{A}^T{)}^T$
$=A^{-1}{A}^{T}A(A^{-1}{)}^T[\therefore {\left(AB\right)}^T=B^T{A}^T]$
$=A^{-1}A{A}^T(A^{-1}{)}^T$
$=I{A}^T(A^{-1}{)}^T[\therefore \left(A^{-1}A\right)=I]$
$=(A^{-1}A{)}^T$
$=I^T=I$