1. ## Matrix Algebra 3

The question is:

Given that dim(m) = 2x2 and M^2 - M + 2I =0, show that M^4 + 3M^2 + 4I =0

It doesn't say what M is, though

2. $\displaystyle M^2-M+2I=0\Rightarrow M^3=M^2-2M$

$\displaystyle (M^2-M+2I)^2=0$

$\displaystyle M^4-2M^3+5M^2-4M+4I=0$

Replace $\displaystyle M^3$:

$\displaystyle M^4-2M^2+4M+5M^2-4M+4I=0\Rightarrow M^4+3M^2+4I=0$

3. Why does M^3 = M^2 - 2M ?

4. You are told that $\displaystyle M^2- M+ 2I= 0$. Adding M- 2I to both sides, $\displaystyle M^2= M- 2I$. Now multiply both sides by M to get $\displaystyle M^3= M^2- 2M$

5. Thank you! But I don't understand the rest of the working either?

Why is (M^2 - M + 2I)^2 =0?

And could you explain the rest of it too ,please?

6. Originally Posted by juliak
Thank you! But I don't understand the rest of the working either?

Why is (M^2 - M + 2I)^2 =0?

And could you explain the rest of it too ,please?
If $\displaystyle A = 0*I$ then $\displaystyle A^2 = A*A = 0*I*0*I = 0*I$

The rest is simply technical.