Question 1

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? Figure from the original question paper Which of the following options are correct?\ Let A and B be square matrices of the same order.