# Finding diagonal matrices with complex entries

Find all diagonal 3 x 3 matrices A with complex entries such that $A^2$=$$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \\ \end{bmatrix}$$

Can anyone please help me how to start this? I'm not sure how I should approach this question. Any help would be greatly appreciated!

• A $3\times 3$ diagonal matrix has the form $$A=\begin{bmatrix} a & 0 & 0\\ 0 & b & 0\\ 0 & 0 &c\end{bmatrix}$$ for some $a,b,c\in\mathbb{C}$. Now what must $A^2$ look like? – Zack Cramer Mar 13 '17 at 4:09
• does that mean that$$A^2=\begin{bmatrix} a^2 & 0 & 0 \\ 0 & b^2 & 0 \\ 0 & 0 & c^2 \\ \end{bmatrix}$$? so does complex entries mean that entries should be in the a+bi form??? – user425030 Mar 13 '17 at 4:14
• Yes, each entry has that form. Now compare this to the matrix $$\begin{bmatrix} 0 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & -1\end{bmatrix}.$$ – Zack Cramer Mar 13 '17 at 4:24
• So I can do, something like (a+bi)^2 = 2 and same with -1? – user425030 Mar 13 '17 at 4:28
• Yup! That should do it. – Zack Cramer Mar 13 '17 at 4:29

In the complex plane, the $n$th root of unity has produces $n$ points on the unit circle. For the square root, $n=2$: $\theta_1 = 0$, $\theta_2 = \pi$. When $z = 2$, $$\sqrt{z} = \pm \sqrt{2}$$