Question

A square martix A is said to be idempotent if A=A^2. The only real numbers that are idempotent are 1 and 0, but this is not true for martices. Find a 2x2 idempotent other than 1 or 0.

Answer 1

Assume the 2x2 matrix to be A= \
a b
c d

Do the matrix multiplication AA and equate each element to A,
a=a²+bc
b=b(a+d)
c=c(a+d)
d=bc+d²

From which we conclude
a+d=1
or
a=1-d
and
bc = a-a² = d-d²

Take a=4, then d=-3
bc=4-4²=-12,
If b=2, c=-6

So
4 2
-6 -3
should be idempotent.
Check my calculations, and calculate the product to ensure that A is idempotent.