Matrix Algebra 3

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August 29th 2009, 12:36 AM
juliak

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
August 29th 2009, 01:17 AM
red_dog

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

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

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

Replace $M^3$ :

$M^4-2M^2+4M+5M^2-4M+4I=0\Rightarrow M^4+3M^2+4I=0$
August 29th 2009, 02:30 AM
juliak

Why does M^3 = M^2 - 2M ?
August 29th 2009, 02:44 AM
HallsofIvy

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 $M^3= M^2- 2M$
August 30th 2009, 12:27 AM
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?
August 30th 2009, 05:34 AM
Defunkt

Quote:

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.

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