Question

Is there a 2x2 matrix 'A' where
AA = A
and A is not entirely composed of 0s?

Answer 1

The identity matrix? Other than that idk

Answer 2

The identity matrix wouldn't work:

I did some hardcore math letting
A = a,b],[c,d
AxA resulted in these equations:
a^2 + bc =a
ab + bd = b
ac + cd = c
bc + d^2 = d

Then b(a+d) = b
so a+d = 1
And I deduced that if -a^2+a = -d^2+d, then a = d so a = 0.5 and d = 0.5
Then bc = 0.25, then I guessed b = 0.25 and c = 0.25

So THIS works:
\[A=\left[\begin{matrix}0.25 & 0.25 \\ 0.25 & 0.25\end{matrix}\right] A^2=\left[\begin{matrix}0.25 & 0.25 \\ 0.25 & 0.25\end{matrix}\right]\]

Answer 3

Proof that the identity matrix wouldn't work:
\[I^{2} = \left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right] \left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right] = \left[\begin{matrix}1*1+0*0 & 1*0+0*1 \\ 0*1+1*0 & 0*0+1*1\end{matrix}\right]=\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]\]

wait a minute it DOES WORK LOL

Answer 4

I'm an idiot, anything times the identity matrix is itself.
AxI = A

Answer 5

is this linear algebra?

Answer 6

i have that in my program...looks like we're going to be best friends :)

Answer 7

Yup, this is linear algebra.

Which chapter are you on?

Answer 8

i haven't started this beast yet but will in the future...

Answer 9

the null matrix too