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? 

From the relation given that $A^2 =-I$
We get an annihilating polynomial $p(x)=x^2+1$
And clearly the minimal polynomial of A divides this and hence the roots are $i$ and $-i$.
Just choose any a,b such that their product is $-1$ and you'll get your matrix..the prod has to be  $-1$ since the product of the eigenvalues is the determinant and for your matrix this comes out to be $-ab$. So $$-ab=1$$
