Question

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

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

Answer 1

The correct option is A a skew-symmetric matrix

Given, $A$ is a skew symmetric matrix. Therefore, $A^T=-A$. $n$ is an odd positive integer.

Now, to check whether $A^n$ is a skew symmetric matrix or not.

$(A^n{)}^T=(A.A^{n-1}{)}^T$

$=(A^{n-1}{)}^T.A^T$

$=(A.A^{n-2}{)}^T.(-A)$

$=(A^{n-2}{)}^T.A^T.(-A)$

$=(A.A^{n-3}{)}^T(-A\left)\right(-A)$

$=(A^{n-3}{)}^T.A^T.(-A{)}^2$ and so on....

$=(-A{)}^n$

$=(-1{)}^n{A}^n$

$=-A^n$

Therefore, $A^n$ is also a skew symmetric matrix if $A$ is a skew symmetric matrix