031 Review Part 3, Problem 5

Find a formula for  ${\begin{bmatrix}1&-6\\2&-6\end{bmatrix}}^{k}$  by diagonalizing the matrix.

Foundations:
Recall:
1. To diagonalize a matrix, you need to know the eigenvalues of the matrix.
2. Diagonalization Theorem
An  $n\times n$  matrix  $A$  is diagonalizable if and only if  $A$  has  $n$  linearly independent eigenvectors.
In fact,  $A=PDP^{-1},$  with  $D$  a diagonal matrix, if and only if the columns of  $P$  are  $n$  linearly
independent eigenvectors of  $A.$  In this case, the diagonal entries of  $D$  are eigenvalues of  $A$  that
correspond, respectively , to the eigenvectors in  $P.$

Solution:

Step 1:

Step 2:

Final Answer:

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