Question

Consider K = 1 −1 1 −1 then we get K2 = 0 Does this hold for real numbers? Motivate.

Answer 1

If ab=0, either a=0 or b=0

-Products of two non-zero numbers is always non-zero

But products of two non-zero matrices can be zero matrix

Using K given above

K=$\begin{bmatrix} 1&-1&;&1&-1 \end{bmatrix}$

That is K$\begin{pmatrix} 1 & -1 \\ 1 & -1 \end{pmatrix}$

K²=(K)(K)

=$\begin{pmatrix} 1 & -1 \\ 1& -1 \end{pmatrix}$ $\begin{pmatrix} 1 & -1\\ 1 & -1 \end{pmatrix}$

$\begin{pmatrix} 1×1+-1×1 & 1×-1+-1×-1\\ 1×1+-1×1 & 1×-1+-1×-1 \end{pmatrix}$

=$\begin{pmatrix} 0 & 0\\ 0& 0 \end{pmatrix}$

=0 (Null matrix)