Question

Determine whether the statement “If $\mathit{A}{\mathit{A}}^{\mathbf{T}}\mathbf{=}{\mathit{A}}^{\mathbf{2}}$for$\mathbf{2}\mathbf{\times }\mathbf{2}$ matrix A then A must be symmetric.

Short answer

The given statement is true.

Step-by-step solution

Definition of a symmetric matrix

“Matrix A is symmetric if $A^T=A$.”

Explanation of the solution

Consider the statement as follows:

“If $A{A}^T=A^2$for $\mathbf{2}\mathbf{\times }\mathbf{2}$matrix A, then A must be symmetric

Simplify the given relation as follows:

$$\begin{gathered} A{A}^T=A^2 \\ A{A}^T-A^2=0 \\ A\left(A^T-A\right)=0 \\ \Rightarrow A=0 or A^T-A=0 \end{gathered}$$

Simplify further as follows:

$$\begin{gathered} A^T-A=0 \\ {A}^T=A \\ A=0 \end{gathered}$$

This implies that A is a symmetric matrix.

Hence, the given statement “If $A{A}^T=A^2$for $2\times 2$matrix A, then A must be symmetric” is true.