Question

If A is a skew-symmetric matrix, then A² is a ___________ matrix.

Answer 1

It is given that, A is a skew-symmetric matrix.

$\therefore A^T=-A$ .....(1)

Now,

${\left(A^2\right)}^T$

$={\left(AA\right)}^T$

$=A^T{A}^T$ [For any matrices X, Y, (XY)^T = Y^TX^T]

$=\left(-A\right)\left(-A\right)$ [Using (1)]

$=A^2$

Since ${\left(A^2\right)}^T=A^2$, so the matrix A² is a symmetric matrix.

If A is a skew-symmetric matrix, then A² is a __symmetric__ matrix.