For any square matrix write whether
Question:

For any square matrix write whether $A A^{\top}$ is symmetric or skew-symmetric.

Solution:

Here,

$\left(A A^{T}\right)^{T}=\left(A^{T}\right)^{T} A^{T} \quad\left[\because(A B)^{T}=B^{T} A^{T}\right]$

$\Rightarrow\left(A A^{T}\right)^{T}=A A^{T} \quad\left[\because\left(A^{T}\right)^{T}=A\right]$

Thus, $A A^{\top}$ is a symmetric matrix.