List of top Mathematics Questions on Linear Algebra asked in GATE MA

Consider the linear system \(Mx = b\), where \(M = \begin{pmatrix} 2 & -1 \\ -4 & 3 \end{pmatrix}\) and \(b = \begin{pmatrix} -2 \\ 5 \end{pmatrix}\).
Suppose \(M = LU\), where L and U are lower triangular and upper triangular square matrices, respectively. Consider the following statements:
P: If each element of the main diagonal of L is 1, then \(\text{trace}(U) = 3\).
Q: For any choice of the initial vector \(x^{(0)}\), the Jacobi iterates \(x^{(k)}, k = 1,2,3,...\) converge to the unique solution of the linear system \(Mx = b\).
Then

Let \(A = [a_{ij}]\) be a \(3 \times 3\) real matrix such that
\[ A \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = 2 \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = 2 \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \quad \text{and} \quad A \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} = 4 \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}. \]
If \(m\) is the degree of the minimal polynomial of \(A\), then \(a_{11} + a_{21} + a_{31} + m\) equals ..................

Let \( M = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & 4 \\ 0 & 0 & 1 \end{bmatrix} \) and \( I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \). If \(6M^{-1} = M^2 - 6M + \alpha I\) for some \(\alpha \in \mathbb{R}\), then the value of \(\alpha\) is equal to ................

Consider \(\mathbb{R}^4\) with the inner product \(\langle x, y \rangle = \sum_{i=1}^4 x_i y_i\), for \(x = (x_1, x_2, x_3, x_4)\) and \(y = (y_1, y_2, y_3, y_4)\).
Let \(M = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = x_3\}\) and \(M^\perp\) denote the orthogonal complement of M. The dimension of \(M^\perp\) is equal to ................

Let \( M = \begin{bmatrix} 4 & -3 \\ 1 & 0 \end{bmatrix} \).
Consider the following statements:
P: \(M^8 + M^{12}\) is diagonalizable.
Q: \(M^7 + M^9\) is diagonalizable.
Which of the following statements is correct?

Let \(T: \mathbb{R}^3 \to \mathbb{R}^3\) be a linear transformation satisfying
\(T(1, 0, 0) = (0, 1, 1)\), \(T(1, 1, 0) = (1, 0, 1)\) and \(T(1, 1, 1) = (1, 1, 2)\).
Then

Let A be a \(3 \times 3\) real matrix with det(\(A + i I\)) = 0, where \(i = \sqrt{-1}\) and I is the \(3 \times 3\) identity matrix. If det(A) = 3, then the trace of \(A^2\) is ..................

Let \(T: \mathbb{R}^4 \to \mathbb{R}^4\) be a linear transformation and the null space of \(T\) be the subspace of \(\mathbb{R}^4\) given by \[ \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : 4x_1 + 3x_2 + 2x_3 + x_4 = 0 \}. \] If \(\text{Rank}(T - 3I) = 3\), where \(I\) is the identity map on \(\mathbb{R}^4\), then the minimal polynomial of \(T\) is