Lunch will be served
A well-known result of elementary algebra states that if a and b are complex numbers, then the quadratic equation
x2 + ax + b = 0
has at most two solutions in complex numbers.
We will consider the same problem where a, b, and x are two-by-two matrices over the complex numbers. We will solve this problem using basic ideas from linear algebra: eigenvectors, eigenvalues and determinants. The solution generalizes to equations of degree m
xm + am-1xm-1 + ... + a1x + a0 = 0
where x and the ai are n-by-n matrices over the complex numbers.