Question

TRUE OR FALSE

The matrix ${\mathit{A}}^{\mathbf{T}}\mathit{A}$ is symmetric for all matrices A.

Short answer

The given statement is true.

Step-by-step solution

Definition of the symmetric matrix

The matrix ${\mathit{A}}^{\mathbf{T}}\mathit{A}$ is symmetric for all matrices A.

Explanation of the solution

Consider the matrix as follows:

$Q=A^{T}A$

Now, taking transpose on both sides and simplify.

$$\begin{gathered} Q^T={\left(A^{T}A\right)}^T \\ {Q}^T=A^T{\left(A^T\right)}^T{A}^T \\ {Q}^T=A^{T}A \end{gathered}$$

Simplify further as follows:

$$\begin{gathered} Q^T=A^{T}A \\ {Q}^T=Q \end{gathered}$$

This implies that Q is a symmetric matrix.

Hence, the given statement that the matrix $A^{T}A$is symmetric for all matrices A” is true.