## Question 6:

### Part (a) — Way 1

| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{A} + \mathbf{B} = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix}$ | M1A1 | M1: Correct attempt at matrix addition with 3 elements correct; A1: Correct matrix |
| $2\mathbf{A} - \mathbf{B} = \begin{pmatrix} 5 & 1 \\ -2 & -1 \end{pmatrix}$ | M1A1 | M1: Correct attempt to double $\mathbf{A}$ and subtract $\mathbf{B}$, 3 elements correct; A1: Correct matrix |
| $(\mathbf{A}+\mathbf{B})(2\mathbf{A}-\mathbf{B}) = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix}\begin{pmatrix} 5 & 1 \\ -2 & -1 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -7 & -2 \end{pmatrix}$ | M1A1 | M1: Correct method to multiply; A1: cao |

**(6 marks total)**

### Part (a) — Way 2

| Answer | Mark | Guidance |
|--------|------|----------|
| $(\mathbf{A}+\mathbf{B})(2\mathbf{A}-\mathbf{B}) = 2\mathbf{A}^2 + 2\mathbf{B}\mathbf{A} - \mathbf{A}\mathbf{B} - \mathbf{B}^2$ | M1A1 | M1: Expands brackets with at least 3 correct terms; A1: Correct expansion |
| $\mathbf{A}^2 = \begin{pmatrix} 3 & 2 \\ -2 & -1 \end{pmatrix}$, $\mathbf{B}\mathbf{A} = \begin{pmatrix} -3 & -1 \\ -1 & 0 \end{pmatrix}$, $\mathbf{A}\mathbf{B} = \begin{pmatrix} -2 & 3 \\ 1 & -1 \end{pmatrix}$, $\mathbf{B}^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ | M1A1 | M1: Attempts $\mathbf{A}^2$, $\mathbf{B}^2$ and $\mathbf{AB}$ or $\mathbf{BA}$; A1: Correct matrices |
| $2\mathbf{A}^2 + 2\mathbf{B}\mathbf{A} - \mathbf{A}\mathbf{B} - \mathbf{B}^2 = \begin{pmatrix} 1 & -1 \\ -7 & -2 \end{pmatrix}$ | M1A1 | M1: Substitutes into their expansion; A1: Correct matrix |

### Part (b) — Way 1

| Answer | Mark | Guidance |
|--------|------|----------|
| $\mathbf{MC} = \mathbf{A} \Rightarrow \mathbf{C} = \mathbf{M}^{-1}\mathbf{A}$ | B1 | May be implied by later work |
| $\mathbf{M}^{-1} = \frac{1}{-2-7}\begin{pmatrix} -2 & 1 \\ 7 & 1 \end{pmatrix}$ | M1 | An attempt at their $\frac{1}{\det\mathbf{M}}\begin{pmatrix} -2 & 1 \\ 7 & 1 \end{pmatrix}$ |
| $\mathbf{C} = \frac{1}{-2-7}\begin{pmatrix} -2 & 1 \\ 7 & 1 \end{pmatrix}\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}$ | dM1 | Correct order required and an attempt to multiply |
| $\mathbf{C} = -\frac{1}{9}\begin{pmatrix} -5 & -2 \\ 13 & 7 \end{pmatrix}$ | A1 | oe |

**(4 marks total; Total: 10)**

### Part (b) — Way 2

| Answer | Mark | Guidance |
|--------|------|----------|
| $\begin{pmatrix} 1 & -1 \\ -7 & -2 \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}$ | B1 | Correct statement |
| $a - c = 2,\ b - d = 1,\ -7a - 2c = -1,\ -7b - 2d = 0$ | M1 | Multiplies correctly to obtain 4 equations |
| $a = \frac{5}{9},\ b = \frac{2}{9},\ c = -\frac{13}{9},\ d = -\frac{7}{9}$ | M1A1 | M1: Solves to obtain values for $a, b, c$ and $d$; A1: Correct values |

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