The image shows a math problem asking to:

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

Proof:

Let $A$ be an arbitrary matrix of size $m \times n$. The product of matrix $A$ and its transpose $A^T$ is given by:

$B = A A^T$

We need to prove that $B$ is a symmetric matrix, i.e., $B^T = B$.

1. Transpose of the product $A A^T$: $B^T = (A A^T)^T$

   Using the property of transposition that $(XY)^T = Y^T X^T$, we get: $B^T = (A^T)^T A^T$

2. Simplification: The transpose of a transpose of a matrix equals the original matrix, i.e., $(A^T)^T = A$. Thus: $B^T = A A^T$

3. Conclusion: Since $B^T = A A^T$ and $B = A A^T$, we conclude that $B = B^T$, which means that $B$ is a symmetric matrix.

Hence, the product of a matrix and its transpose is always symmetric.

Would you like more details or have any questions?

Follow-up questions:

1. Can you provide an example matrix to see this property in action?
2. What is the result if $A$ is already a symmetric matrix?
3. How does this property change if $A$ is not a square matrix?
4. Can this property be extended to complex matrices, and how would that change the proof?
5. Does the size of the matrix $A$ affect the symmetry of the product?

Tip: When dealing with matrix properties, always pay attention to whether matrices are square, as many important properties like symmetry depend on matrix dimensions.