Q9Single correct3 Marks31 Aug 2025

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation 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?

A

S1 is true and S2 is false.

B

S1 is false and S2 is true.

C

Both S1 and S2 are true.

D

Both S1 and S2 are false.