# Paper 1, Section I, A

Let $A$ be a real $3 \times 3$ matrix.

(i) For $B=R_{1} A$ with

$R_{1}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta_{1} & -\sin \theta_{1} \\ 0 & \sin \theta_{1} & \cos \theta_{1} \end{array}\right)$

find an angle $\theta_{1}$ so that the element $b_{31}=0$, where $b_{i j}$ denotes the $i j^{\text {th }}$entry of the matrix $B$.

(ii) For $C=R_{2} B$ with $b_{31}=0$ and

$R_{2}=\left(\begin{array}{ccc} \cos \theta_{2} & -\sin \theta_{2} & 0 \\ \sin \theta_{2} & \cos \theta_{2} & 0 \\ 0 & 0 & 1 \end{array}\right)$

show that $c_{31}=0$ and find an angle $\theta_{2}$ so that $c_{21}=0$.

(iii) For $D=R_{3} C$ with $c_{31}=c_{21}=0$ and

$R_{3}=\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \theta_{3} & -\sin \theta_{3} \\ 0 & \sin \theta_{3} & \cos \theta_{3} \end{array}\right)$

show that $d_{31}=d_{21}=0$ and find an angle $\theta_{3}$ so that $d_{32}=0$.

(iv) Deduce that any real $3 \times 3$ matrix can be written as a product of an orthogonal matrix and an upper triangular matrix.