Question

If $A$ is a square matrix, then

• A. $A{A}^T$ is symmetric matrix and $A^{T}A$ is skew-symmetric matrix.
• B. $A{A}^T$ is skew-symmetric matrix and $A^{T}A$ is symmetric matrix.
• C. Both $A{A}^T$ and $A^{T}A$ are symmetric matrices.
• D. Both $A{A}^T$ and $A^{T}A$ are skew-symmetric matrices.

Answer 1

The correct option is C Both $A{A}^T$ and $A^{T}A$ are symmetric matrices.

$(A{A}^T{)}^T=(\left(A^T{)}^T{A}^T\right)=A{A}^T\Rightarrow A{A}^T\text{symmetric matrix}.(A^{T}A{)}^T=(A^T\left(A^T{)}^T\right)=A^{T}A\Rightarrow A^{T}A\text{symmetric matrix}.$