Question:
If $A$ is a skew-symmetric matrix, then $A^{2}$ is a________ matrix.
Solution:
It is given that, $A$ is a skew-symmetric matrix.
$\therefore A^{T}=-A$
Now,
$\left(A^{2}\right)^{T}$
$=(A A)^{T}$
$=A^{T} A^{T}$ [For any matrices $X, Y,(X Y)^{\top}=Y^{\top} X^{\top}$ ]
$=(-A)(-A)$ [Using (1)]
Since $\left(A^{2}\right)^{T}=A^{2}$, so the matrix $A^{2}$ is a symmetric matrix.
If $A$ is a skew-symmetric matrix, then $A^{2}$ is a symmetric matrix