CAIE A-Level Further Math 1.4.1 Matrix operations

Practise matrix arithmetic with square and non-square matrices, including products, powers and parameter values found from entries.

[Maximum number: 5]

Let k be a constant. The matrices A, B and C are given by

\mathbf{A}=\left(\begin{array}{lll} 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rr} 0 & -2 \\ -1 & 3 \\ 0 & 0 \end{array}\right) \quad \text { and } \quad \mathbf{C}=\left(\begin{array}{rrr} -2 & -1 & 1 \\ 1 & 1 & 3 \end{array}\right)

It is given that A is singular.

Show that $\mathbf{C A B}=\left(\begin{array}{rr}3 & -7 \\ -9 & 3\end{array}\right)$ .

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

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

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1.4.1—Matrix operations question 1