Question

Let A be a matrix. Show that the matrix is symmetric. [Hint: Show that this matrix $\mathit{A}{\mathit{A}}^{\mathbf{t}}$ equals its transpose with the help of Exercise 17b.]

Short answer

Therefore, it is shown that $\mathit{A}{\mathit{A}}^{\mathbf{t}}$is symmetric.

Step-by-step solution

Property of transpose of matrix

Given: A is a matrix.

To prove:$\mathit{A}{\mathit{A}}^{\mathbf{t}}$ is symmetric

Now, we can use$\mathbf{(}\mathit{A}\mathit{B}{\mathbf{)}}^{\mathbf{t}}\mathbf{=}{\mathit{B}}^{\mathbf{t}}{\mathit{A}}^{\mathbf{t}}$ and$\mathbf{(}{\mathit{A}}^{\mathbf{t}}{\mathbf{)}}^{\mathbf{t}}\mathbf{=}\mathit{A}$

$\mathbf{(}\mathit{A}{\mathit{A}}^{\mathbf{t}}{\mathbf{)}}^{\mathbf{t}}\mathbf{=}\mathbf{(}{\mathit{A}}^{\mathbf{t}}{\mathbf{)}}^{\mathbf{t}}{\mathit{A}}^{\mathbf{t}}\mathbf{=}\mathit{A}{\mathit{A}}^{\mathbf{t}}$

Symmetric matrix

Since, the transposed of the matrix$\mathit{A}{\mathit{A}}^{\mathbf{t}}$ results in the matrix$\mathit{A}{\mathit{A}}^{\mathbf{t}}$ itself, $\mathit{A}{\mathit{A}}^{\mathbf{t}}$is symmetric.