+0

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

0
117
4

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

Guest Jul 18, 2017
edited by Guest  Jul 18, 2017
Sort:

#1
+26236
+2

Like so:

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

Alan  Jul 18, 2017
#2
0

Now I feel dumb for not noticing they simply replaced Awith a already known value. Thank you.

Guest Jul 18, 2017
#3
+26236
+2

No need to feel dumb; hindsight is a wonderful thing!

Alan  Jul 18, 2017
#4
+18610
+2

If A2 = 2A + 3I, find A3 and A4 in linear form kA + sI

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

heureka  Jul 18, 2017

### 29 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details