Mathematics - Matrices Question with Solution | TestHub

For (P): We check the transpose of $A{A}^T$. $$(A{A}^T{)}^T=(A^T{)}^T{A}^T=A{A}^T$$ Thus, $A{A}^T$ is always a symmetric matrix. For (Q): We check the transpose of $A-A^T$. $$(A-A^T{)}^T=A^T-(A^T{)}^T=A^T-A=-(A-A^T)$$ Thus, $A-A^T$ is always a skew-symmetric matrix. For (R): If $A$ is an orthogonal matrix, then $A{A}^T=I$. We need to check if $A^T$ is orthogonal. $$(A^T)(A^T{)}^T=A^{T}A$$ Since $A{A}^T=I$, it implies $A^T=A^{-1}$. Substituting this, $A^{T}A=A^{-1}A=I$. Thus, $A^T$ is an orthogonal matrix. For (S): If $A$ is a symmetric matrix, then $A=A^T$. We need to check the transpose of $A^{-1}$. $$(A^{-1}{)}^T=(A^T{)}^{-1}$$ Since $A=A^T$, we have $(A^T{)}^{-1}=A^{-1}$. Therefore, $(A^{-1}{)}^T=A^{-1}$, which means $A^{-1}$ is a symmetric matrix.