Question

A is a 2x2 matrix

Let A = {(1) (1/d) (c) (d)} Find the numbers c and d such that A^2 =0

Answer 1

$$A=\begin{pmatrix} 1& 1/d \\ c & d \end{pmatrix}$$

$$A^2=\begin{pmatrix} 1& 1/d \\ c & d \end{pmatrix}\begin{pmatrix} 1& 1/d \\ c & d \end{pmatrix}$$

$$=\begin{pmatrix} 1+c/d& 1+1/d \\ c+cd & c/d+d^2 \end{pmatrix}=\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$$

$$\begin{matrix} 1+c/d=0\\ 1+1/d=0\\ c+c/d=0\\ c/d+d^2=0\\ \end{matrix}$$

$$c=1, d=-1$$

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

$$A^2=\begin{pmatrix} 1& -1 \\ 1 & -1 \end{pmatrix}\begin{pmatrix} 1& -1 \\ 1 & -1 \end{pmatrix}=\begin{pmatrix} 1-1& -1+1 \\ 1-1 & -1+1 \end{pmatrix}$$

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