IAL 2026 Jan FP1 Q5

Question

[!problem]

The matrix $A$ is given by

$A = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2} \end{pmatrix}$

(a) Describe fully the single geometric transformation represented by the matrix $A$.

(2)

The matrix $B$ represents an enlargement with centre $(0, 0)$ and scale factor $2\sqrt{2}$.

(b) Write down matrix $B$.

(1)

Given that

• $I$ is the identity matrix
• the transformation represented by $(A + I)$ followed by the transformation represented by $B$ is represented by the matrix $C$
• the triangle $T$ is transformed by $C$ to the triangle $T'$
• $T'$ has an area of $12 + \sqrt{2}$

(c) determine, in simplest form, the exact area of $T$.

(4)