A2 June 2024 Paper 2 Q3

OCR ACurrent spec7 marksMatrices

3 Matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 4 & -3 \\ -2 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 3 & -5 \\ 0 & 1 \end{pmatrix}\).

(a) Find \(2\mathbf{A} - 4\mathbf{B}\). [2]

(b) Write down the matrix \(\mathbf{C}\) such that \(\mathbf{AC} = 2\mathbf{A}\). [1]

(d) In this question you must show detailed reasoning.
Use \(\mathbf{A}^{-1}\) to solve the equations \(4x - 3y = 7\) and \(-2x + 2y = 9\). [3]

Mark scheme (a)

Scheme	Marks	AO
\(\mathbf{C} = 2\begin{pmatrix} 4 & -3 \\ -2 & 2 \end{pmatrix} - 4\begin{pmatrix} 3 & -5 \\ 0 & 1 \end{pmatrix}\) \(= \begin{pmatrix} 8 & -6 \\ -4 & 4 \end{pmatrix} - \begin{pmatrix} 12 & -20 \\ 0 & 4 \end{pmatrix}\) \(= \begin{pmatrix} 8 - 12 & -6 - -20 \\ -4 - 0 & 4 - 4 \end{pmatrix}\)	M1	1.1
\(= \begin{pmatrix} -4 & 14 \\ -4 & 0 \end{pmatrix}\)	A1	1.1
	[2]

Notes

M1: Sufficient working to demonstrate knowledge of scalar multiplication of a matrix and subtraction of matrices. Can be implied by 3 out of 4 entries correct.

A1: Or BC

Scheme	Marks	AO
\((\mathbf{C} =)\ \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\) or \(2\mathbf{I}\)	B1	2.2a
	[1]

Scheme	Marks	AO
\((\det\mathbf{A} = 4 \times 2 - (-3)(-2) = 8 - 6 =)\ 2\)	B1	1.1
	[1]

Mark scheme (d)

Scheme	Marks	AO
\(\begin{pmatrix} 4 & -3 \\ -2 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 \\ 9 \end{pmatrix}\)	M1	1.1
\(\dfrac{1}{2}\begin{pmatrix} 2 & 3 \\ 2 & 4 \end{pmatrix}\begin{pmatrix} 7 \\ 9 \end{pmatrix}\) or \(\mathbf{A}^{-1}\begin{pmatrix} 7 \\ 9 \end{pmatrix}\) if \(\mathbf{A}^{-1}\) defined	M1	1.1
so \(x = \dfrac{41}{2},\ y = 25\)	A1	1.1
	[3]

Notes

M1: DR Expressing the system in matrix form. Can be implied by the next line. Matrix method must be used. Any other method 0/3.

M1: Forming correct solution as matrix/vector product with inverse matrix. FT their \(\det\mathbf{A}\) from (c). Except for this, inverse must be correct.

A1: \(20\frac{1}{2}\) or 20.5
Condone \(\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \frac{41}{2} \\ 25 \end{pmatrix}\) but not \(\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{2}\begin{pmatrix} 41 \\ 50 \end{pmatrix}\)