Let\( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear transformation defined by \(T(\mathbf{x}) = A\mathbf{x} \) , where A is a \(3 \times 3 \) matrix given by:
\(A = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \\ \end{bmatrix} \)
Which of the following statements is true about T ?
1
T is injective but not surjective.
2
The eigenvalues of T are 1, -2, and 3 .
3
T is an isometry.
4
The determinant of T is 216.