(a) Let $A$ be an $n × n$ real matrix such that $(A + I)^4 = 0$ where $I$ denotes the identity matrix. Show that $A$ is non-singular. (b) Give an example of a nonzero $2×2$ real matrix $A$ such that $x′Ax = 0$ for all real vectors $x$.
(a) Note that $(A + I)^4 = 0=>A^4+4A^3+6A^2+4A+I=0$ Then I am stuck.
(b)$\begin{pmatrix} 0 & 1\\-1 & 0 \end{pmatrix}$ is an example. I get the matrix from intuitive idea. Is there another way to get such matrix.