A final exam problem

mark

Our short final exam will be this Friday, December 6 at 11:30 for the MW folks and next Tuesday, December 10 at 8:00 AM for the TR folks.

There will be three problems:

1. Very much like the problem below,
2. Very much like one of the UTMOST end of semester questions, and
3. Something else.

---

Here's the problem below: Let
$$
A=\left(
\begin{array}{ccc}
-6 & 5 & 1 \\
-10 & 9 & 5 \\
0 & 0 & 2
\end{array}
\right).
$$

a. Diagonalize $A$.
b. Compute
$$A^{100} \left(\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}\right).$$

These parts might not be unrelated.

Student23

I am a little confused on how to go about solving part b. I was able to diagonalize the matrix, but now I am not sure what to do. Does anyone have any suggestions?

Student02

@Student23 said:
I am a little confused on how to go about solving part b. I was able to diagonalize the matrix, but now I am not sure what to do. Does anyone have any suggestions?

We proved earlier that $A^{n} = S^{-1}D^{n}S$, so I think we can just find $A^{100}$ by finding $S^{-1}D^{100}S$. $D^{100}$ is way easier to compute than $A^{100}$...

Student23

@Student02 said:

@Student23 said:
I am a little confused on how to go about solving part b. I was able to diagonalize the matrix, but now I am not sure what to do. Does anyone have any suggestions?

We proved earlier that $A^{n} = S^{-1}D^{n}S$, so I think we can just find $A^{100}$ by finding $S^{-1}D^{100}S$. $D^{100}$ is way easier to compute than $A^{100}$...

Thank you so much! This really helped clear things up!