AS June 2019 Paper 1 Q2
2 Matrices \(\mathbf{P}\) and \(\mathbf{Q}\) are given by \(\mathbf{P} = \begin{pmatrix} 1 & k & 0 \\ -2 & 1 & 3 \end{pmatrix}\) and \(\mathbf{Q} = \begin{pmatrix} (1 + k) & -1 \end{pmatrix}\) where \(k\) is a constant.
Exactly one of statements A and B is true.
Statement A: \(\mathbf{P}\) and \(\mathbf{Q}\) (in that order) are conformable for multiplication.
Statement B: \(\mathbf{Q}\) and \(\mathbf{P}\) (in that order) are conformable for multiplication.
| Scheme | Marks | AO |
|---|---|---|
| B | B1 | 2.2a |
| (For matrices to be conformable for multiplication) the number of columns of the first must equal the number of rows in the second oe “the number of rows of P is equal to the number of columns of Q” | E1 | 1.2 |
| [2] |
Notes
B1: Note “B” is that QP is conformable
E1: Statement can be general or specific. Allow eg \((1 \times 2) \times (2 \times 3) = (1 \times 3)\) provided that it is clear which two numbers must be the same
Since told exactly one is true it is sufficient to give a reason why one is true or why one is false
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{QP} = \begin{pmatrix} (1 + k) & -1 \end{pmatrix}\begin{pmatrix} 1 & k & 0 \\ -2 & 1 & 3 \end{pmatrix}\) \(= \begin{pmatrix} (1 + k) + 2 & k(1 + k) - 1 & -3 \end{pmatrix}\) | M1 | 1.1 |
| \(\begin{pmatrix} (k + 3) & (k^2 + k - 1) & -3 \end{pmatrix}\) | A1 | 1.1 |
| [2] |
Notes
M1: Correct method for multiplying matrices (can be implied by any one entry correct)
If PQ attempted then M0A0 unless explicitly rejected
A1: Accept un-simplified elements