Maths 2 End Term: 31 August 2025, Set QDF1 (May 2025 term)

Let $T_1, T_2 : \mathbb{R}^2 \to \mathbb{R}^2$ be two linear transformations defined as:

$T_1(x,y) = (3x - y, 2x - y), \quad T_2(x,y) = (2x + \alpha y, \alpha x)$

Let $A$ and $B$ be the matrix representations of $T_1$ and $T_2$ with respect to the standard bases of $\mathbb{R}^2$, respectively.

Consider the following statements:

S1: Considering the basis $\{e_1 + e_2, -e_2\}$, we can see that if $\alpha = 1$, $A$ and $B$ are similar matrices.

S2: If $\alpha = 2$, $A$ and $B$ are similar matrices.

Which of the following options is true?

Which of the following options are correct?
Let A and B be square matrices of the same order.