## Question 8:

### Part (a):
$A^{-1} = \frac{1}{3}\begin{pmatrix} 3 & 0 \\ -2 & 1 \end{pmatrix}$ | **B1** | No guidance notes

### Part (b):
$[AB =]\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 2 & -3 \end{pmatrix}$ | **M1** (1.1) | Correctly finding either **AB** or **BA**

$[BA =]\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -2 & -3 \end{pmatrix}$ so they are not the same | **A1** (2.2a) | Must give correct calculation and correct conclusion. $AB \neq BA$ is fine for conclusion. General statement "matrices not commutable" not OK unless linked to particular case.

### Part (c):
The matrix representing R is $C(BA)$... | **M1** (3.1a) | Or the matrix representing S is $(CB)A$. Must have $R=C(BA)$ or $S=(CB)A$ with brackets or equivalent correct. Need to see the brackets or equivalent. BOD sight of $T_A$

...and the matrix representing S is $(CB)A$ and by associativity of matrix multiplication, $C(BA) = (CB)A$ (so R and S are the same) | **A1** (3.2a) | Correct form for S and either explicit equality or associativity property explicitly mentioned. M1 A0 only if state that matrices are commutative. If **M0** then **SC B1** for observation that $(AB)C = A(BC)$ or any other correct statement of associativity of matrix multiplication.

### Part (d)(i):
$\det A = 3$ or $\det B = -1$ | **M1** (1.1) | Question says "Determine" so answer only is 0

$\det(CBA) = -3(a^2+1) < 0$ (since $a$ is real) [so the orientation is reversed]. Order is $D'$, $G'$, $F'$, $E'$ | **A1** (2.2a) | Some justification must be given but condone incorrect order of matrices. Orientation reversed is fine, but must see correct $\det(CBA)$. Can consider $\det(C)\det(B)\det(A)$ instead. Allow "the order is clockwise". **SC1** If neither det **A** or det **B** explicitly calculated then allow B1 for det **A**, det **C** positive and det **B** negative so orientation reversed. **SC2** If only det (C) considered (i.e. candidate thinks transformation R is represented by **C**) then allow **B1** for $a^2+1>0$ so orientation is the same.

### Part (d)(ii):
$15(a^2+1)$ | **B1FT** (1.1) | FT $5\times$ their determinant from (i) if found. FT only if their determinant is negative.

### Part (d)(iii):
The determinant is not zero... | **M1** (1.1) | Understanding that the matrix associated with the transformation is non-singular. Allow M1 for "the determinant is negative". Or equating det to 0 and solving even if conclusion wrong. Can be considering det **C** or det **R** here. Allow M1 for "determinant is positive" ONLY if it is clear where the determinant has come from (e.g. from det **C**).

...so the inverse transformation exists. | **A1FT** (2.2a) | Condone use of "matrix" rather than "transformation". Allow follow through if using det **C**. If have found matrix singular when $a = i$ then need to discount this as not real. If their determinant is 0 then **SC1** only can be awarded for showing understanding that the transformation associated with a singular matrix does not have an inverse. Need to be showing non-zero determinant.

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