Q5Single correct5 Marks21 Dec 2025

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be the linear transformation defined by $T(x, y, z) = (3x + y + z, ax + bz, x - y)$ and $A$ be the matrix representation of $T$ with respect to the standard ordered basis $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$ for both the domain and the co-domain. Consider the following statements. S1: If $a = 1, b = 0$, then $A$ is similar to $B = \begin{bmatrix} 2 & 0 & 0 \\ 1 & -1 & 0 \\ 2 & 1 & 3 \end{bmatrix}$. S2: If $a = 0, b = -1$, then $A$ is equivalent to $B = \begin{bmatrix} 1 & -1 & 0 \\ 4 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Choose the correct option.

A

Both the statements S1 and S2 are true.

B

Only statement S1 is true.

C

Only statement S2 is true.

D

Both the statements are false.