Question

If A is a skew-symmetric matrix and n is an even positive integer, then $A^n$ is

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

Answer 1

The correct option is A a symmetric matrix

Since, A is a skew-symmetric matrix.
$A^T=-A$
$(A^2{)}^T=(AA{)}^T=A^T{A}^T=(-A)(-A)=A^2$
$\Rightarrow (A^2{)}^T=A^2$
Hence, $A^2$ is symmetric.
$(A^4{)}^T=(A^2{A}^2{)}^T=\left(A^2{)}^T\right(A^2{)}^T=A^2{A}^2=A^4$
$\Rightarrow (A^4{)}^T=A^4$
Hence, $A^4$ is symmetric.
Hence, $A^n$ is symmetric for n to be positive even integer.