If , then show that

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

If $A = (\begin{array}{cc} p & q \\ 0 & 1 \end{array})$ , then show that $A^{8} = (\begin{array}{cc} p^{8} & q (\frac{p^{8} - 1}{p - 1}) \\ 0 & 1 \end{array})$

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$A^{2} = [\begin{array}{cc} p & q \\ 0 & 1 \end{array}] [\begin{array}{cc} p & q \\ 0 & 1 \end{array}] = [\begin{array}{cc} p^{2} + 0 & pq + q \\ 0 & 1 \end{array}]$

A^{3} = [\begin{array}{cc} p^{2} & q (p + 1) \\ 0 & 1 \end{array}] [\begin{array}{cc} p & q \\ 0 & 1 \end{array}] = [\begin{array}{cc} p^{3} & p^{2} q + q (pq) \\ 0 & 1 \end{array}] = [\begin{array}{cc} p^{3} & q (p^{2} + p + 1) \\ 0 & 1 \end{array}]

-------- and so on

A^{8} = [\begin{array}{cc} p^{8} & q \frac{p^{8} - 1}{p - 1} \\ 0 & 1 \end{array}]

Hence proved

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