1. ## Consider this matrix

consider the matrix M= (2 0, 0 2)^n, for n>1 ; n E 1.

Calculate M^n for n= 1, 2, 3, 4, 5, 10, 20, 50. describe any pattern observed, and generalize the pattern into an expression for the matrix M^n in terms of n.

Determine detM for the powers of the matrices calculated above. Describe any pattern observed, generalize the pattern into an expression for det(M^n) in terms of n.

~Zet

2. This might help:

$\begin{pmatrix}
{2}&{0}\\
{0}&{2}
\end{pmatrix}^n=2^n\begin{pmatrix}
{1}&{0}\\
{0}&{1}
\end{pmatrix}^n=\begin{pmatrix}
{2^n}&{0}\\
{0}&{2^n}
\end{pmatrix}$

As for the determinant:

$\begin{vmatrix}
{2^n}&{0}\\
{0}&{2^n}
\end{vmatrix}=(2^n)^2=2^{2n}$