You're told , so
So equating the like elements gives
.
Now try to equate and .
The matrix A = [[a, i][i, b]], where i^2 = -1, a = (1/2)(1 + sqrt(5)) and
b = (1/2)(1 - sqrt(5)) has the property that A^2 = A. Describe all 2 X 2 matrices A with complex entries such that A^2 = A.
I don't know why they give us the matrix A, as in, I don't know how to use it to solve the problem. would a method involving something like this work?
[[a, b][c, d]] [[a, b][c, d]] = [[a, b][c, d]]
and try to solve for a, b, c, d?
BTW, the answer is this:
[[a, b][c, 1-a]], where b and c are arbitrary and a is any solution of the quadratic equation a^2 - a + bc = 0.
Well, it obviously isn't the only matrix that satisfies since the identity matrix also does.
I don't think there is a reason they give you the matrix ; it's just used as motivation so that you get a chance to see that there even exist matrices other than the identity. Another example is the matrix .
And yes, your original proposed method is exactly what you should do, although it is more efficient to note and solve for the equation