Maths 2 End Term: 31 August 2025, Set QDF3 (May 2025 term)

Consider a linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^3$ defined as:

$T(x,y,z) = (x + y - z, ay + 2bz, x - y + z)$

Let $A$ be the matrix representation of $T$ with respect to the standard bases of $\mathbb{R}^3$ and let $B = \begin{bmatrix} 3 & 0 & 3 \\ 1 & 0 & 1 \\ 1 & 0 & 3 \end{bmatrix}$.

Consider the following statements:

S1: If $A$ and $B$ are similar, then the range of the linear transformation $T$ is a plane in $\mathbb{R}^3$.

S2: If $a = 2$ and $b = -1$, then $A$ and $B$ are equivalent matrices, but not similar.

Which of the following options is true?

Which of the following options are correct?