Question

a.Give an example of a (nonzero) skew-symmetric 3×3 matrix A, and compute.

b.If an n×nmatrix Ais skew-symmetric, is matrix ${\mathit{A}}^{\mathbf{2}}$necessarily skew-symmetric as well? Or is ${\mathit{A}}^{\mathbf{2}}$necessarily symmetric?

Short answer

$A^2$need not be skew-symmetric, but it will be symmetric.

Step-by-step solution

skew-symmetric matrices.

Consider an m×nmatrix A.

The transpose${\mathit{A}}^{\mathbf{T}}$of Ais the$\mathit{n}\mathbf{\times }\mathit{m}$matrix whoseijth entry is jiththe entry of A: The roles of rows and columns are reversed.

It says that a square matrix Ais symmetricif, ${\mathit{A}}^{\mathbf{T}}\mathbf{=}\mathit{A}$and Ais called skew-symmetricif ${\mathit{A}}^{\mathbf{T}}\mathbf{=}\mathit{A}$.

For the part (a) here it needs a skew-symmetric matrix. So, according to the definition of skew-symmetric matrix the diagonal entries will be zero.

Consider the matrix,

$A=\left[\begin{array}{ccc}0& 1& 1\\ -1& 0& 1\\ -1& -1& 0\end{array}\right]$

Clearly, the matrix A is skew-symmetric matrix. Now

$$\begin{gathered} A^2=\left[\begin{array}{ccc}0& 1& 1\\ -1& 0& 1\\ -1& -1& 0\end{array}\right]\times \left[\begin{array}{ccc}0& 1& 1\\ -1& 0& 1\\ -1& -1& 0\end{array}\right] \\ =\left[\begin{array}{ccc}0& -1& 1\\ -1& -2& -1\\ 1& -1& -2\end{array}\right] \end{gathered}$$

Consider the part (b).

Let A be ann×n skew-symmetric matrix. $A^2$need not to be skew-symmetric, the example taken in the part (a) shows this.

For $A^2$is symmetric or not, observe the following terms.

$$\begin{gathered} {\left(A^2\right)}^T={\left(AA\right)}^T \\ =A^T{A}^T \\ ={\left(A^T\right)}^2 \\ ={\left(-A\right)}^2 \\ ={\left(-1\right)}^2{A}^2 \\ =A^2 \end{gathered}$$

Thus, the matrix $A^2$is always symmetric.

Hence, $A^2$will be symmetric.