CAIE A-Level Further Mathematics 1.4.3 Inverse Matrix Products

Practise inverting a product of non-singular matrices by reversing the factor order, including transformation matrices assembled from two successive mappings.

Exam points

find each component inverse and write the inverse product in reversed factor order
use the reversed product to obtain the matrix that maps the final image back to its source

[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.

B1

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

B1

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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1.4.3—Inverse matrix products question 1