FP1 June 2012 Q2

EdexcelOld spec6 marksMatrices

(a) Given that \[\mathbf{A} = \begin{pmatrix} 3 & 1 & 3 \\ 4 & 5 & 5 \end{pmatrix} \quad \text{and} \quad \mathbf{B} = \begin{pmatrix} 1 & 1 \\ 1 & 2 \\ 0 & -1 \end{pmatrix}\] find \(\mathbf{AB}\). (2)

(b) Given that \[\mathbf{C} = \begin{pmatrix} 3 & 2 \\ 8 & 6 \end{pmatrix}, \quad \mathbf{D} = \begin{pmatrix} 5 & 2k \\ 4 & k \end{pmatrix}, \text{ where } k \text{ is a constant}\] and \[\mathbf{E} = \mathbf{C} + \mathbf{D}\] find the value of \(k\) for which \(\mathbf{E}\) has no inverse. (4)

Mark scheme (a)

Scheme	Marks
\(\mathbf{A} = \begin{pmatrix} 3 & 1 & 3 \\ 4 & 5 & 5 \end{pmatrix},\ \mathbf{B} = \begin{pmatrix} 1 & 1 \\ 1 & 2 \\ 0 & -1 \end{pmatrix}\)
\(\mathbf{AB} = \begin{pmatrix} 3 & 1 & 3 \\ 4 & 5 & 5 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & 2 \\ 0 & -1 \end{pmatrix}\)
\(= \begin{pmatrix} 3 + 1 + 0 & 3 + 2 - 3 \\ 4 + 5 + 0 & 4 + 10 - 5 \end{pmatrix}\) A correct method to multiply out two matrices. Can be implied by two out of four correct (unsimplified) elements in a dimensionally correct matrix. A 2x2 matrix with a number or a calculation at each corner.	M1
\(= \begin{pmatrix} 4 & 2 \\ 9 & 9 \end{pmatrix}\) Correct answer	A1
	[2]

Notes

A correct answer with no working can score both marks

Mark scheme (b)

Scheme	Marks
\(\mathbf{C} = \begin{pmatrix} 3 & 2 \\ 8 & 6 \end{pmatrix},\ \mathbf{D} = \begin{pmatrix} 5 & 2k \\ 4 & k \end{pmatrix}\), where \(k\) is a constant,
\(\mathbf{C} + \mathbf{D} = \begin{pmatrix} 3 & 2 \\ 8 & 6 \end{pmatrix} + \begin{pmatrix} 5 & 2k \\ 4 & k \end{pmatrix} = \begin{pmatrix} 8 & 2k + 2 \\ 12 & 6 + k \end{pmatrix}\) An attempt to add C to D. Can be implied by two out of four correct (unsimplified) elements in a dimensionally correct matrix.	M1
\(\mathbf{E}\) does not have an inverse \(\Rightarrow \det\mathbf{E} = 0\).
\(8(6 + k) - 12(2k + 2)\) Applies “\(ad - bc\)” to \(\mathbf{E}\) where \(\mathbf{E}\) is a 2x2 matrix.	M1
\(8(6 + k) - 12(2k + 2) = 0\) States or applies \(\det(\mathbf{E}) = 0\) where \(\det(\mathbf{E}) = ad - bc\) or \(ad + bc\) only and \(\mathbf{E}\) is a 2x2 matrix.	M1
\(48 + 8k = 24k + 24\) \(24 = 16k\)
\(k = \tfrac{3}{2}\)	A1 oe
	[4]
	6 marks

Notes

Note \(8(6 + k) - 12(2k + 2) = 0\) or \(8(6 + k) = 12(2k + 2)\) could score both M’s