# Matrix Algebra 3

• Aug 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
• Aug 29th 2009, 01:17 AM
red_dog
\$\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\$
• Aug 29th 2009, 02:30 AM
juliak
Why does M^3 = M^2 - 2M ?
• Aug 29th 2009, 02:44 AM
HallsofIvy
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\$
• Aug 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?
• Aug 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 \$\displaystyle A = 0*I\$ then \$\displaystyle A^2 = A*A = 0*I*0*I = 0*I\$

The rest is simply technical.