For A=beginpmatrix2&a\\0&1endpmatrix it can be shown that A^n=beginpmatrix2^n&(2^n-1)a\\0&1endpmatrix for every positive integer n. State the top right entry of A^5 when a=3.

Paper 2 · November 2012 · Mathematical induction with matrices

For $A=\begin{pmatrix}2&a\\0&1\end{pmatrix}$ it can be shown that $A^{n}=\begin{pmatrix}2^{n}&\left(2^{n}-1\right)a\\0&1\end{pmatrix}$ for every positive integer $n$ . State the top right entry of $A^{5}$ when $a=3$ .

Model answer

Also accepted: $93$

Explanation

The top right entry is

(2^{n}-1)a

. With

n=5

and

a=3

this is

(32-1)(3)=31\times3=93

Derived from ZIMSEC Mathematics Paper 2, November 2012, Q1

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