Given matrix $A = \begin{pmatrix} 2 && 2 \\ 2 && 2\end{pmatrix}$, I want to find two square roots of A.
I have to go about this with only very introductory-type tools, those covered in an introductory matrix operations chapter.
My Approach
Since I know that the square root matrix is a 2x2 matrix, let the square root matrix be $B = \begin{pmatrix} a && b \\ c && d\end{pmatrix}$.
Now, for B to be a square root matrix of A, the following must hold true: BB = A. Evaluating BB I get $BB = \begin{pmatrix} a^2 + bc && b(a + d) \\ c(a+d) && d^2 + bc\end{pmatrix}$
This leaves me with the following equations:
$a^2 + bc=2$
$d^2 + bc=2$
$b(a+d)=2$
$c(a+d)=2$
From here on I've tried solving the equations but none of my attempts yielded the correct solution. I get the feeling I'm overlooking something very basic.