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How do I multiply a $2 \times 2$ matrix by the power of $i$ in a summation formula?

$$\left[ \begin {array}{cc} a&b\\ c&d\end {array} \right] ^{i}$$

I want to get the values of the first row, first column, and second row, first column. So I require two summation formulas. My Maple program is broken and doesn't like algebraic expressions in matrices. I can get the first values, but I can't get anything above them.

Is this where the gamma thing comes into play?

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  • $\begingroup$ Why was this downvoted? Was it because "My Maple program is broken and doesn't like algebraic expressions in matrices."? $\endgroup$ – Pedro Tamaroff Mar 30 '13 at 1:07
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Let $A$ be your matrix. You have the basic matrix multiplication formula: $$ \left( \begin{array}{c c} a_{1 1} & a_{1 2} \\ a_{2 1} & a_{2 2} \end{array} \right) \cdot \left( \begin{array}{c c} b_{1 1} & b_{1 2} \\ b_{2 1} & b_{2 2} \end{array} \right) = \left(\begin{array}{c c} a_{1 1} b_{1 1} + a_{1 2} b_{2 1} & a_{1 1} b_{1 2} + a_{1 2} b_{2 2} \\ a_{2 1} b_{1 1} + a_{2 2} b_{2 1} & a_{2 1} b_{1 2} + a_{2 2} b_{2 2} \end{array}\right) $$ Using that, you can set up recurrences for $a_{i j}^{(k)}$, the elements of $A^k$. If you are lucky with the $a_{i j}$, the recurrences turn out nice.

Another way is to find some matrix $P$ such that $P^{-1} D P = A$, where $D$ is a diagonal matrix ($d_{1 2} = d_{2 1} = 0$), for which powers are simple to compute (check with the above multiplication formula): $$ \left(\begin{matrix} d_{1 1} & 0 \\ 0 & d_{2 2} \end{matrix}\right)^k = \left(\begin{matrix} d_{1 1}^k & 0 \\ 0 & d_{2 2}^k \end{matrix}\right) $$ With that representation, $A^k = P^{-1} D^k P$. Look up diagonalizable matrices.

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