June 2017 Paper 2 Q2
2 \(\mathbf{A} = \begin{pmatrix} 4 & -1 \\ 3 & -2 \end{pmatrix} \qquad \mathbf{B} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
(a) Work out \(\quad \mathbf{A}^2\) [2 marks]
(b) \(k\mathbf{B} = \begin{pmatrix} 11 - 3k \\ 11 - 6k \end{pmatrix} \quad\) where \(k\) is a constant.
Work out the value of \(k\). [2 marks]
(c) Give a reason why it is not possible to work out \(\mathbf{BA}\) [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(\begin{pmatrix} 13 & -2 \\ 6 & 1 \end{pmatrix}\) | B2 | B1 13 or \(-2\) or 6 or 1 in correct position in a 2 by 2 matrix |
Additional guidance
| Condone missing brackets for B2 or B1 if numbers in a 2 by 2 array | |
| Brackets may be square or curly etc | |
| Ignore commas and fraction lines | |
| \(\begin{pmatrix} 13 & -2 \\ 6 & 1 \end{pmatrix}\) followed by further work | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(5k = 11 - 3k\) or \(2k = 11 - 6k\) or \(11 - 3k = \dfrac{5}{2}(11 - 6k)\) or \(8k = 11\) | M1 | oe Any one correct equation |
| \(\dfrac{11}{8}\) or \(1\dfrac{3}{8}\) or 1.375 with no incorrect equation seen | A1 | oe |
Additional guidance
| \(\begin{pmatrix} 5k \\ 2k \end{pmatrix} = \begin{pmatrix} 11 - 3k \\ 11 - 6k \end{pmatrix}\) with no further correct work | M0 |
| Ignore subsequent attempt to convert \(\dfrac{11}{8}\) to a mixed fraction or decimal | M1A1 |
| Ignore subsequent attempt to convert \(1\dfrac{3}{8}\) to an improper fraction or decimal | M1A1 |
| Ignore subsequent rounding or truncation of 1.375 | M1A1 |
| Answer only 1.37 or 1.38 or 1.4 | M0 |
| T & I is 2 or zero |
| Answer | Mark | Comments |
|---|---|---|
| Valid explanation | B1 | eg Number of columns of B does not equal number of rows of A |
Additional guidance
| ‘2 by 1’ (or ‘2 × 1’) matrix means 2 rows and 1 column | |
| ‘First matrix’ means B and ‘second matrix’ means A | |
| B columns ≠ A rows | B1 |
| B rows ≠ A columns | B0 |
| B columns ≠ A columns | B0 |
| B rows ≠ A rows | B0 |
| Number of rows in second matrix cannot be more than number of columns in first matrix | B1 |
| B has 1 column, A has 2 rows | B1 |
| A should only have 1 row | B1 |
| A has too many rows | B1 |
| B only has one column | B1 |
| B needs another column | B1 |
| It is a 2 x 1 multiplied by a 2 × 2 | B1 |
| There’s nothing to multiply the 3 by | B1 |
| It is a 2 x 2 multiplied by 2 × 1 | B0 |
| It is a 1 by 2 multiplied by a 2 by 2 | B0 |
| B values can’t multiply with all the A values | B0 |
| They are not compatible | B0 |
| Because the dimensions of A and B are different | B0 |
| Can’t work it out this way round | B0 |
| Can work out AB but not BA | B0 |
| B has to be a 2 by 2 matrix | B0 |