Question

Let A be a square matrix, then $A{A}^T$ and $A^{T}A$ are

• A. Non-symmetric rectangular matrices
• B. Symmetric and non-identical square matrices
• C. Non-symmetric square matrices
• D. Symmetric and identical square matrices

Answer 1

Answer: (B)

Symmetric and non-identical square matrices

We have,
$(A{A}^T{)}^T=(\left(A^T{)}^T{A}^T\right)$ [By reversal law]
$=A{A}^T$ [$\because (A^T{)}^T=A$]
Therefore, $A{A}^T$ is symmetric.
Also $A{A}^T$ and $A^{T}A$ are non-identical matrix as we know a matrix and its transpose are not same.