Question

If A is a symmetric matrix and $nϵN$ then $A^n$ is

• A. symmetric
• B. skew-symmetric
• C. a diagonal matrix
• D. none of these

Answer 1

The correct option is A symmetric

Since, A is symmetric
$A^T=A$
Now, $(A^2{)}^T=(AA{)}^T=A^T{A}^T=A^2$
i.e.$(A^2{)}^T=A^2$
Hence, $A^2$ is symmetric.
$(A^3{)}^T=(A^{2}A{)}^T=A^T(A^2{)}^T=A{A}^2=A^3$
i.e.$(A^3{)}^T=A^3$
Hence, $A^3$ is symmetric.
Hence, for all $n\in N$, $A^n$ is symmetric