CAIE A-Level Further Math ASematics 1.4.2 Determinants and Inverses

[Maximum number: 5]

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.

Find, in terms of k, the value of $\tan \theta$ for which M-I is singular.

$\mathbf{M}-\mathbf{I}=\left(\begin{array}{cc}\cos 2 \theta-1 k \cos 2 \theta-\sin 2 \theta \sin 2 \theta k \sin 2 \theta+\cos 2 \theta-1\end{array}\right)$ $(\cos 2 \theta-1)(k \sin 2 \theta+\cos 2 \theta-1)-k \sin 2 \theta \cos 2 \theta+\sin ^{2} 2 \theta[=0]$
Evaluates $\operatorname{det}(\mathbf{M}-\mathbf{I})$ $2-2 \cos 2 \theta-k \sin 2 \theta=0$
Brackets removed correctly and =0

$4 \sin ^{2} \theta=2 k \sin \theta \cos \theta$
Uses $1-\cos 2 \theta=2 \sin ^{2} \theta$ and $\sin 2 \theta=2 \sin \theta \cos \theta$ or all
necessary double angle formulae.

$\tan \theta=\frac{1}{2} k$
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CAIE A-Level Further Math AS 1.4.2 Determinants and Inverses Questions

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CAIE A-Level Further Math AS 1.4.2 Determinants and Inverses Questions question 1