CAIE A-Level Further Math AS 1.4.3 Inverse Matrix Products Questions

Practise expressing inverse matrices as products and using matrices for transformations.

Exam points

express inverse matrices as products
calculate inverses of matrix powers
use matrices to represent geometric transformations

[Maximum number: 2]

Let $\mathbf{A}=\left(\begin{array}{ll}3 & 0 \\ 1 & 1\end{array}\right)$ .

Find, in terms of n, the inverse of $\mathbf{A}^{n}$ .

$\operatorname{det} \mathbf{A}^{n}=\operatorname{det}\left(\begin{array}{cc}3^{n} & 0 \frac{1}{2}\left(3^{n}-1\right) & 1\end{array}\right)=3^{n}$

Or a multiple of $\mathbf{A}^{-n}=2^{-1} 3^{-n}\left(\begin{array}{cc}2 & 0 1-3^{n} & 2 \times 3^{n}\end{array}\right)$ seen.
$\mathbf{A}^{-n}=3^{-n}\left(\begin{array}{cc}1 \frac{1}{2}\left(1-3^{n}\right) 3^{n}\end{array}\right)$
OE $\mathbf{A}^{-n}=2^{-1} 3^{-n}\left(\begin{array}{cc}2 1-3^{n} 2 \times 3^{n}\end{array}\right)$

2

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CAIE A-Level Further Math AS 1.4.3 Inverse Matrix Products Questions question 1