CAIE A-Level Further Math ASematics 1.4.3 Inverse Matrix Products

[Maximum number: 2]

The matrix M is given by $\mathbf{M}=\left(\begin{array}{rr}\cos 2 \theta & -\sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta\end{array}\right)\left(\begin{array}{ll}1 & k \\ 0 & 1\end{array}\right)$ , where $0<\theta<\pi$ and k is a non-zero constant. The matrix M represents a sequence of two geometrical transformations, one of which is a shear.

Write $\mathbf{M}^{-1}$ as the product of two matrices, neither of which is I.

$\left(\begin{array}{cc}1 k 0 1\end{array}\right)^{-1}=\left(\begin{array}{cc}1 -k 0 1\end{array}\right),\left(\begin{array}{cc}\cos 2 \theta -\sin 2 \theta \sin 2 \theta \cos 2 \theta\end{array}\right)^{-1}=\left(\begin{array}{cc}\cos 2 \theta \sin 2 \theta -\sin 2 \theta \cos 2 \theta\end{array}\right)$ $\mathbf{M}^{-1}=\left(\begin{array}{cc}1 -k 0 1\end{array}\right)\left(\begin{array}{cc}\cos 2 \theta \sin 2 \theta -\sin 2 \theta \cos 2 \theta\end{array}\right)$
Correct order

2

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

Exam points

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