CSIR NET Maths previous year question papers

Let V be the $\mathbb{R}-$vector space of 5 × 5 real matrices. Let S = {AB − BA | A, B ∈ V} and W denote the subspace of V spanned by S. Let $T : V \to \mathbb{R}$ be the linear transformation mapping a matrix A to its trace. Which of the following statements is true?

Let A, B be 2 × 2 matrices with real entries, and M = AB − BA. Let $I_{2}$ denote the 2 × 2 identity matrix. Which of the following statements are necessarily true?

Let $M_{2}(\mathbb{R})$ denote the $\mathbb{R}-$vector space of 2 × 2 matrices with real entries. Let A = [[1, 2], [0, 3]] and B = [[−1, 0], [1, 5]]. Define a linear transformation $T : M_{2}(\mathbb{R}) \to M_{2}(\mathbb{R})$ by T(X) = AXBᵗ, where Bᵗ denotes the transpose of the matrix B. Which of the following statements are true?