The image shows the following mathematical problem:

"g) Prove that the product of a matrix and its transpose is a symmetric matrix."

Proof:

Let $A$ be an arbitrary matrix, and we want to show that the product $A A^T$ is a symmetric matrix.

A matrix $B$ is said to be symmetric if $B = B^T$. So, we need to prove that $(A A^T)^T = A A^T$.

1. Transpose of a Product: One property of the transpose operation is that $(A B)^T = B^T A^T$ for any two matrices $A$ and $B$. Applying this to the product $A A^T$, we get: $(A A^T)^T = (A^T)^T A^T$

2. Transpose of the Transpose: Another property is that the transpose of a transpose is the original matrix, i.e., $(A^T)^T = A$. So we have: $(A A^T)^T = A A^T$

Since $(A A^T)^T = A A^T$, we have shown that the product $A A^T$ is a symmetric matrix.

Conclusion:

The product of any matrix and its transpose is always a symmetric matrix.

Do you need more details on any of the steps? Here are some related questions that might interest you:

1. What is the definition of a transpose of a matrix?
2. Can the product $A^T A$ (instead of $A A^T$) also be symmetric?
3. What are the conditions for a matrix to be symmetric?
4. How does the transpose