Question

If A and B are arbitrary ${\mathit{n}}^{\mathbf{\times }}\mathit{n}$matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

${\mathit{A}}^{\mathbf{T}}\mathit{A}$.

Short answer

$A^{T}A$ is symmetric.

Step-by-step solution

Condition to be a symmetric.

A matrix is symmetric if and only if it is equal to its transpose.

All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal. A matrix is skew-symmetric if and only if it is the opposite of its transpose.

Example

For example,

Let $A=\left(\begin{array}{ccc}1& 1& -1\\ 1& 2& 0\\ -1& 0& 5\end{array}\right)$.

Thus, it gives

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

Thus, If A and B are arbitrary $n\times n$ matrices then the matrices$A^{T}A$ is symmetric.

Hence, the matrices is symmetric.