Question

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

• A. An idempotent matrix
• B. An involutory matrix
• C. An orthogonal matrix
• D. None of these

Answer 1

The correct option is C 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, $B=A^{-1}{A}^T$
$B^{-1}=(A^{-1}{A}^T{)}^{-1}$
$=\left(A^T{)}^{-1}\right(A^{-1}{)}^{-1}$
$=(A^{-1}{)}^{-1}A$
$=\left(A\right)A=A^2$
$\Rightarrow B^{-1}=B^T$
Hence, $B$ is orthogonal.