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
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$\displaystyle M^2-M+2I=0\Rightarrow M^3=M^2-2M$ $\displaystyle (M^2-M+2I)^2=0$ $\displaystyle M^4-2M^3+5M^2-4M+4I=0$ Replace $\displaystyle M^3$: $\displaystyle M^4-2M^2+4M+5M^2-4M+4I=0\Rightarrow M^4+3M^2+4I=0$
Why does M^3 = M^2 - 2M ?
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 $\displaystyle M^3= M^2- 2M$
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?
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.
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