Matrix Algebra 3

**juliak** · August 29th 2009, 12:36 AM

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

**red_dog** · August 29th 2009, 01:17 AM

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

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

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

Replace $M^3$ :

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

**juliak** · August 29th 2009, 02:30 AM

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

**HallsofIvy** · August 29th 2009, 02:44 AM

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

**juliak** · August 30th 2009, 12:27 AM

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?

**Defunkt** · August 30th 2009, 05:34 AM

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 $A = 0*I$ then $A^2 = A*A = 0*I*0*I = 0*I$

The rest is simply technical.

Thread: Matrix Algebra 3