Question

TRUE OR FALSE? The equation $\mathit{d}\mathit{e}\mathit{t}\left(A^T\right)\mathbf{=}\mathit{d}\mathit{e}\mathit{t}\left(A\right)$holds for all $\mathbf{2}\mathbf{\times }\mathbf{2}$matrices A.

Short answer

The given statement is true.

Step-by-step solution

 Step 1: Definition of the determinant.

Suppose $\mathit{A}\mathbf{=}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$is a $\mathbf{2}\mathbf{\times }\mathbf{2}$matrix.

Then the determinant of matrix Ais defined as,

$$\begin{gathered} \mathit{d}\mathit{e}\mathit{t}\left(A\right)\mathbf{=}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right] \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathit{a}\mathit{d}\mathbf{-}\mathit{b}\mathit{c} \end{gathered}$$

 Step 2: Check whether the given statement is a true or false.

The given statement is the equation $det{\left(A\right)}^T=det\left(A\right)$holds for all $2\times 2$matrix A.

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$.

Then,

$$\begin{gathered} A^T={\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^T \\ {A}^T=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right] \end{gathered}$$

Find $det\left(A^T\right),det\left(A\right)$

$$\begin{gathered} det\left(A\right)=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right| \\ =ad-bc \\ det\left(A^T\right)=\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|c \\ =ad-cd \end{gathered}$$

This is clear that, the equation $det\left(A^T\right)=det\left(A\right)$is holdsfor all$2\times 2$ matrix A

Then the given statement is true.