If A^{2} = 2A + 3I, find A^{3 }and A^{4 }in linear form kA + sI

I = "eye" as in identity matrix.

This is an example queston with given steps to obtain the answer. I just don't understand why they did what they did. Here are the steps:

1. A^{3 }= A * A^{2}

2. A(2A + 3I)

3. 2A^{2} + 3Al

From this part on, I'm confused to what is going on.

4. 2(2A + 3I) + 3AI (Why does step 3 translate to this?)

5. 7A + 6I (HUH?)

And then it starts all over again with A^{4} = A * A^{3 }

I'm trying to self study matrix. Step 4 and 5 is confusing. Someone care to explain what is going on? Thanks!

Edit: Some errors in the step

Guest Jul 18, 2017

edited by
Guest
Jul 18, 2017

#1**+2 **

Like so:

(Note that any matrix multiplied by the identity matrix stays as itself.)

Alan Jul 18, 2017

#4**+2 **

**If A2 = 2A + 3I, find A ^{3} and A^{4} in linear form kA + sI **

\(\begin{array}{|lrcll|} \hline 1.& A^3 &=& A * A^2 \quad & | \quad A^2 = 2A + 3I \\ 2.& A^3 &=& A *(2A + 3I) \\ 3.& A^3 &=& 2A^2 + 3AI \quad & \text{ or } \quad 3AI =A^3-2A^2\\\\ 4.& A^3 &=& \underbrace{2A^2}_{=2(2A + 3I)} + \underbrace{A^3 - 2A^2}_{=3AI} \\ & A^3 &=& 2(2A + 3I) + 3AI \quad & | \quad A*I = A \\ & A^3 &=& 2(2A + 3I) + 3A \\ & A^3 &=& 4A + 6I + 3A \\ 5. & A^3 &=& 7A + 6I \\ \hline \end{array}\)

heureka Jul 18, 2017