Question

Is the determinant of a transpose the same?

Answer 1

Explanation of why the determinant of the matrix and the determinant of the transpose of a matrix is the same:

Let's consider an example;

$A\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

The determinant of $A$ is;

$$\begin{gathered} \left|A\right|=\left|\begin{array}{cc}1& 2\\ 3& 4\end{array}\right| \\ =4-6 \\ =-2 \end{gathered}$$

The transpose of a matrix is found by interchanging its rows into column or columns into rows.

The transpose of the matrix $A$ is;

$A^T=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$

The determinant of the transpose matrix of $A$ is;

$$\begin{gathered} \left|A^T\right|=\left|\begin{array}{cc}1& 3\\ 2& 4\end{array}\right| \\ =4-6 \\ =-2 \end{gathered}$$

Hence proved that $\left|A\right|=\left|A^T\right|$

Therefore, the determinant of the matrix and the determinant of the transpose of a matrix is the same.