Question

If $A$ is a non singular matrix such that $A{A}^T=A^{T}A=I$, and $B=A^{-1}{A}^T$, then matrix $B$ is

• A. an involuntary matrix
• B. an orthogonal matrix
• C. an idempotent matrix
• D. None of these

Answer 1

The correct option is B an orthogonal matrix

Given: $A{A}^T=A^{T}A=I$

$\Rightarrow A$ is an orthogonal matrix.
$\Rightarrow A^{-1}=A^T$
Now, given $B=A^{-1}{A}^T$
$B^T=(A^{-1}{A}^T{)}^T$
$=A(A^{-1}{)}^T$
$=A(A^T{)}^T$
$=AA=A^2$
Again, $B=A^{-1}{A}^T$
$B^{-1}=(A^{-1}{A}^T{)}^{-1}$
$=\left(A^T{)}^{-1}\right(A^{-1}{)}^{-1}$
$=(A^T{)}^{-1}A$
$=(A^{-1}{)}^{T}A$
$=(A^T{)}^{T}A$
$=A^2$

$\Rightarrow B^{-1}=B^T$

Hence, $B$ is also orthogonal.

Hence, option B is correct.