Question

Let $A$ and $B$ be two $n\times n$ real symmetric matrices. Which of the following is/are correct?

• A. If $A$ is an involuntry matrix then, $A^{-1}=A$
• B. If $A$ is an orthogonal matrix then, $A^{-1}=A$
• C. The product $AB$ is symmetric if $AB=BA$
• D. If the product $AB$ is diagonal then, $AB=BA$

Answer 1

Answer: (A), (B), (C), (D)

If the product $AB$ is diagonal then, $AB=BA$

(i) $A^2=I$
$\Rightarrow A.A=I$$\because A$ is a symmetric matrix
$\Rightarrow A=A^{-1}(A$ is involuntry then, $A^{-1}=A$$)$

(ii) $A{A}^T=I$
$\Rightarrow A.A=I$
$\Rightarrow A^{-1}=A(A$ is orthogonal then, $A^{-1}=A$$)$

(iii) $(AB{)}^T=AB$ $\because A\&B$ are symmetric matrices
$\Rightarrow (AB{)}^T=B^T{A}^T=BA$
If $BA=AB$ then, $(AB{)}^T=AB$
$(AB$ is symmetric if $AB=BA)$

(iv) Let product of $A$ and $B$ be $\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$
$\Rightarrow (AB{)}^T=AB{B}^T{A}^T=BA\Rightarrow BA=AB$

$(AB$ is diagonal matrix, then $AB=BA)$