Find a $2 \times 2$ matrix $A$ with each main diagonal entries $0$, and with $A^2 = -I.$ I'm not sure how to tackle this problem. 
I'm not certain what is meant by "each main diagonal entries 0". Does this mean:
$$A=\begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$$
I'm also not sure what it means by $A^2 = -I$? 
From Wikipedia, I gathered that the identity matrix is an $n \times n$ square matrix with one's on the main diagonal and zero's elsewhere. What would would be considered the main diagonal? 

The main diagonal runs from the top left to the bottom right, so you're looking for a matrix like
$$A = \left[ \begin{array}{cc} 0 & a \\ b & 0 \end{array}\right]$$
such that $A^2 = -I$. This can be done by trial and error, or by writing a system of a few equations after finding what $A^2$ is.
