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# Need help with matrixes. Find A^3 and A^4 in linear form.

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If A2 = 2A + 3I, find Aand Ain 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= A * A2

2. A(2A + 3I)

3. 2A2 + 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 A4 = A * A

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

Jul 18, 2017
edited by Guest  Jul 18, 2017

#1
+27374
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Like so:

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

Jul 18, 2017
#2
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Now I feel dumb for not noticing they simply replaced Awith a already known value. Thank you.

Guest Jul 18, 2017
#3
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No need to feel dumb; hindsight is a wonderful thing!

Alan  Jul 18, 2017
#4
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If A2 = 2A + 3I, find A3 and A4 in linear form kA + sI

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

Jul 18, 2017