Question

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

• A. $I$
• B. $-I$
• C. $A$
• D. $B$

Answer 1

The correct option is A $I$

Given, $A{A}^T=A^{T}A$ and $B=A^{-1}{A}^T$
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 $B.$
$B{B}^T=\left(A^{-1}{A}^T\right)(A^{-1}{A}^T{)}^T$
$=A^{-1}{A}^{T}A(A^{-1}{)}^T[\because {\left(AB\right)}^T=B^T{A}^T]$
$=A^{-1}A{A}^T(A^{-1}{)}^T$
$=I{A}^T(A^{-1}{)}^T[\because \left(A^{-1}A\right)=I]$
$=(A^{-1}A{)}^T$
$=I^T=I$