CAIE A-Level Further Math 1.4 Matrices Question Bank

$\left|\begin{array}{ll}1 2 5\end{array}\right|-k\left|\begin{array}{ll}2 3 5\end{array}\right|+3\left|\begin{array}{ll}2 3 2\end{array}\right|=0 \Rightarrow-1-k+3=0 \Rightarrow k=2$

M1 A1

Sets determinant of A equal to zero.

$\left(\begin{array}{ccc}-2 -1 1 1 3\end{array}\right)\left(\begin{array}{ccc}1 2 2 1 3 2 5\end{array}\right)\left(\begin{array}{cc}0 -2 -1 0 0\end{array}\right)=\left(\begin{array}{ccc}-2 -1 1 1 3\end{array}\right)\left(\begin{array}{cc}-2 -1 -1 -2 0\end{array}\right)$

M1

Multiplying two matrices correctly, correct
dimensions.

$\left(\begin{array}{cc}3 -7 -9 3\end{array}\right)$

M1 A1

Completing matrix multiplication, AG.

5

$\left(\begin{array}{cc}3 -7 -9 3\end{array}\right)\binom{x}{y}=\binom{3 x-7 y}{-9 x+3 y}$

B1

Transforms $\binom{x}{y}$ to $\binom{X}{Y}$.

-9 x+3 m x=m(3 x-7 m x)

M1 A1

Uses y=m x and Y=m X.

$-9+3 m=3 m-7 m^{2} \Rightarrow 7 m^{2}=9$

A1

$y=\frac{3}{\sqrt{7}} x$ and $y=-\frac{3}{\sqrt{7}} x$

A1

5

$\mathbf{D}=\left(\begin{array}{ll}\alpha 0 \alpha\end{array}\right)$

B1

$\mathbf{E}=\left(\begin{array}{ll}\beta 0 1\end{array}\right)$

B1

$\mathbf{F}=\left(\begin{array}{ll}0 1 0\end{array}\right)$

B1

$\left(\begin{array}{cc}3 -7 -9 3\end{array}\right)=\left(\begin{array}{ll}\alpha 0 \alpha\end{array}\right)-9\left(\begin{array}{ll}0 \beta 1 0\end{array}\right)$

M1

Setting up simultaneous equations using their D and
E.

$\mathbf{D}=\left(\begin{array}{ll}3 0 3\end{array}\right) \quad \mathbf{E}=\left(\begin{array}{ll}\frac{7}{9} 0 1\end{array}\right)$

A1

Condone $\alpha=3, \beta=\frac{7}{9}$ if it is clear that they refer to
the correct matrices.

5