IB Maths AI HL Ahl 1 14 Hl Matrices Questions

Practise adding and multiplying matrices, using determinants and inverses to solve systems, and interpreting matrix models for transformations, coding and recurrences.

[Maximum number: 12]

François is a video game designer. He designs his games to take place in two dimensions, relative to an origin O . In one of his games, an object travels on a straight line $L_{1}$ with vector equation

r=\binom{-1}{1}+\lambda\binom{2}{-1}

(a)

Given that T=R X,

[ 12 ]

(i)

use your answers to part (c) to find matrix X.

[ 6 ]

attempt to multiply matrices from part (c) (in any order)

\boldsymbol{X}=\begin{pmatrix}5\sqrt{2}&0\\0&5\sqrt{2}\end{pmatrix}\begin{pmatrix}\cos\left(\frac{\pi}{4}\right)&-\sin\left(\frac{\pi}{4}\right)\\\sin\left(\frac{\pi}{4}\right)&\cos\left(\frac{\pi}{4}\right)\end{pmatrix}

\boldsymbol{X}=\begin{pmatrix}5&-5\\5&5\end{pmatrix}

(ii)

hence, find the value of $\alpha$ .

[ 6 ]

substituting $\boldsymbol{T}$ , $\boldsymbol{R}$ and $\boldsymbol{X}$

\begin{pmatrix}1&7\\7&-1\end{pmatrix}=\begin{pmatrix}\cos 2\alpha&\sin 2\alpha\\\sin 2\alpha&-\cos 2\alpha\end{pmatrix}\begin{pmatrix}5&-5\\5&5\end{pmatrix}

multiplying by inverse (in any order)

\begin{pmatrix}1&7\\7&-1\end{pmatrix}\begin{pmatrix}5&-5\\5&5\end{pmatrix}^{-1}=\begin{pmatrix}\cos 2\alpha&\sin 2\alpha\\\sin 2\alpha&-\cos 2\alpha\end{pmatrix}

\begin{pmatrix}\cos 2\alpha&\sin 2\alpha\\\sin 2\alpha&-\cos 2\alpha\end{pmatrix}=\begin{pmatrix}-\frac{3}{5}&\frac{4}{5}\\\frac{4}{5}&\frac{3}{5}\end{pmatrix}

\cos 2\alpha=-\frac{3}{5}\quad\text{and}\quad\sin 2\alpha=\frac{4}{5}

\alpha=1.11(=1.107148\ldots)\quad\text{OR}\quad63.4^{\circ}(63.4349\ldots^{\circ})

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IB Maths AI HL Ahl 1 14 Hl Matrices Questions

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IB Maths AI HL Ahl 1 14 Hl Matrices Questions question 1

Question (a)

Question (i)

Question (ii)