Question

If $A^{{}^{\prime }}$ is the transpose of a square matrix A, then

• A. $\left|A\right|\ne \left|A^{{}^{\prime }}\right|$
• B. $\left|A\right|=\left|A^{{}^{\prime }}\right|$
• C. $\left|A\right|+\left|A^{{}^{\prime }}\right|=0$
• D. $\left|A\right|=\left|A^{{}^{\prime }}\right|$ only when A is symmetric

Answer 1

Answer: (B)

$\left|A\right|=\left|A^{{}^{\prime }}\right|$

Let $A=\left[\begin{array}{cc}2& 3\\ 4& 6\end{array}\right]$
$\Rightarrow A^{\prime }=\left[\begin{array}{cc}2& 4\\ 3& 6\end{array}\right]$
Since, determinant of $A=|A|=12-12=0$
and
determinant of $A^{\prime }=|A^{\prime }|=12-12=0$
$\therefore |A|=|A^{\prime }|$
Option B is correct.