1. Matrices

Didn't know where to put this question, but apologies if I had picked the wrong place to put this in!

Find all 2x2 matrices A for which A^2 = A.
They gave me a hint which is to let A be $\displaystyle \begin{bmatrix} a & b \\c & d\end{bmatrix}$

$\displaystyle \begin{bmatrix} a & b \\c & d\end{bmatrix}$ $\displaystyle \begin{bmatrix} a & b \\c & d\end{bmatrix}$ gave me

$\displaystyle \begin{bmatrix} a^2+bc & ab+bd \\ac+cd & bc+d^2\end{bmatrix}$
but how is this A?

2. Re: Matrices

Right off hand, the zero and identity matrices will have this property. That's at least two. What do they have in common? You probably aren't ready to understand the answer, but keep the question in mind until you are.

That will give you four simultaneous equations, not linear. Try solving them. You should wind up with eight possibilities, work through them.

3. Re: Matrices

If

$\displaystyle \begin{bmatrix} a^2+bc & ab+bd \\ac+cd & bc+d^2\end{bmatrix} = \begin{bmatrix} a & b \\c & d\end{bmatrix}$

Then
$\displaystyle a^2+bc=a$
$\displaystyle ab+bd=b$
$\displaystyle ac+cd=c$
$\displaystyle bc+d^2=d$

Try to solve these equations for a, b, c, d. As Zhandele said the identity matrix and the zero matrix will be two results.