A2 June 2022 Paper 1 Q2

OCR ACurrent spec9 marksMatrices

2 The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & -2 \\ 1 & 3 \end{pmatrix}\).

(a) Calculate \(\det\mathbf{A}\). [1]

(b) Write down \(\mathbf{A}^{-1}\). [1]

(c) Hence solve the equation \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}\). [2]

(d) Write down the matrix \(\mathbf{B}\) such that \(\mathbf{AB} = 4\mathbf{I}\). [1]

Matrices \(\mathbf{C}\) and \(\mathbf{D}\) are given by \(\mathbf{C} = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}\) and \(\mathbf{D} = \begin{pmatrix} 0 & 2 & p \end{pmatrix}\) where \(p\) is a constant.

(e) Find, in terms of \(p\),

the matrix \(\mathbf{CD}\)
the matrix \(\mathbf{DC}\).

[3]

It is observed that \(\mathbf{CD} \neq \mathbf{DC}\).

(f) The result that \(\mathbf{CD} \neq \mathbf{DC}\) is a counter example to the claim that matrix multiplication has a particular property. Name this property. [1]

Scheme	Marks	AO
\(\det\mathbf{A} = 3 \times 2 - -2 \times 1 = 8\)	B1	1.1
	[1]

Scheme	Marks	AO
\(\dfrac{1}{8}\begin{pmatrix} 3 & 2 \\ -1 & 2 \end{pmatrix}\)	B1	1.1
	[1]

Mark scheme (c)

Scheme	Marks	AO
\(\begin{pmatrix} x \\ y \end{pmatrix} = \mathbf{A}^{-1}\begin{pmatrix} -1 \\ 2 \end{pmatrix}\)	M1	1.1
\(\Rightarrow x = \dfrac{1}{8},\ y = \dfrac{5}{8}\)	A1	1.1
	[2]

Notes

M1: Sight of their \(\mathbf{A}^{-1}\) multiplied by \(\begin{pmatrix} -1 \\ 2 \end{pmatrix}\)

A1: ft their \(\mathbf{A}^{-1}\). Could be given as a vector.

Scheme	Marks	AO
\(\dfrac{1}{2}\begin{pmatrix} 3 & 2 \\ -1 & 2 \end{pmatrix}\) oe	B1	2.2a
	[1]

Mark scheme (e)

Scheme	Marks	AO
\(\mathbf{DC} = \begin{pmatrix} p \end{pmatrix}\)	B1	1.1
\(\mathbf{CD} = \begin{pmatrix} 0 & 4 & 2p \\ 0 & 0 & 0 \\ 0 & 2 & p \end{pmatrix}\)	M1 A1	1.1 1.1
	[3]

Notes

B1: Must be a matrix; do not award for just \(p\).

M1: for \(3 \times 3\) matrix with at least one correct row or (non-0) column.

SC B2 for correct matrices, but CD and DC interchanged or unspecified
SC B1 for one correct matrix.