A2 June 2025 Paper 2 Q7

AQACurrent spec3 marksMatrices

7 The matrix \(\mathbf{A}\) is defined by \(\mathbf{A} = \begin{bmatrix} 4 & -2 \\ -6 & 3 \end{bmatrix}\)

Find a non-zero \(2 \times 2\) matrix \(\mathbf{B}\) such that \(\mathbf{AB} = 0\) [3 marks]

Mark scheme

Scheme	Marks	AO
Defines elements of \(\mathbf{B}\) (PI) and writes or uses the matrix equation \(\mathbf{AB} = 0\) PI	M1	3.1a
Obtains \(c = 2a\) or \(d = 2b\) OE PI by correct answer.	M1	1.1a
Deduces a matrix of the form \(\begin{bmatrix} n & m \\ 2n & 2m \end{bmatrix}\) with at least one of \(n\) and \(m\) not equal to zero.	A1	2.2a
	(3 marks)

Typical solution

\[\text{Let } \mathbf{B} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]\[\mathbf{AB} = \begin{bmatrix} 4 & -2 \\ -6 & 3 \end{bmatrix}\begin{bmatrix} a & b \\ c & d \end{bmatrix}\]\[= \begin{bmatrix} 4a - 2c & 4b - 2d \\ -6a + 3c & -6b + 3d \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\]\[4a - 2c = 0 \Rightarrow c = 2a\]\[4b - 2d = 0 \Rightarrow d = 2b\]\[\mathbf{B} = \begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}\]