List of top Linear Algebra Questions on Matrix Representation of Linear Transformations asked in GATE MA

Let \(T: \mathbb{R}^3 \to \mathbb{R}^3\) be the linear transformation which reflects every vector in \(\mathbb{R}^3\) through a two-dimensional subspace of \(\mathbb{R}^3\). Let \(P \in \mathbb{R}^{3 \times 3}\) be the matrix representation of \(T\) using the basis

\[ \left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right\}. \]

Then the value of \(2 \times \text{trace}(P) - 3 \times \text{determinant}(P)\) is equal to ______. (answer in integer)

Let \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation defined by \[ T((1, 2)) = (1, 0) \quad \text{and} \quad T((2, 1)) = (1, 1). \] For \( p, q \in \mathbb{R} \), let \( T^{-1}((p, q)) = (x, y) \). Which of the following statements is TRUE?