Question

The product of a matrix and its transpose is an identity matrix. The value of determinant of this matrix is

• A. $-1$
• B. $0$
• C. $\pm 1$
• D. $1$

Answer 1

Answer: (C)

$\pm 1$

Let $A=\left[\begin{array}{ccc}\pm 1& 0& 0\\ 0& \pm 1& 0\\ 0& 0& \pm 1\end{array}\right]$
Thus its transpose will be $A^T=\left[\begin{array}{ccc}\pm 1& 0& 0\\ 0& \pm 1& 0\\ 0& 0& \pm 1\end{array}\right]$.
Therefore, $A.A^T=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
$\Rightarrow \left|A\right|=\pm 1$