List of top Linear Algebra Questions asked in GATE MA

Consider P: Let \( M \in \mathbb{R}^{m \times n} \) with \( m>n \geq 2 \). If \( \text{rank}(M) = n \), then the system of linear equations \( Mx = 0 \) has \( x = 0 \) as the only solution. Q: Let \( E \in \mathbb{R}^{n \times n}, n \geq 2 \) be a non-zero matrix such that \( E^3 = 0 \). Then \( I + E^2 \) is a singular matrix. Which of the following statements is TRUE?

Suppose that the characteristic equation of \( M \in \mathbb{C}^{3 \times 3} \) is \[ \lambda^3 + \alpha \lambda^2 + \beta \lambda - 1 = 0, \] where \( \alpha, \beta \in \mathbb{C} \) with \( \alpha + \beta \neq 0 \).
Which of the following statements is TRUE?
Consider the linear system of equations \( Ax = b \) with
\[ A = \begin{pmatrix} 3 & 1 & 1 \\ 1 & 4 & 1 \\ 2 & 0 & 3 \end{pmatrix}, \quad b = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}. \] Which of the following statements are TRUE?
Consider the following conditions on two proper non-zero ideals \( J_1 \) and \( J_2 \) of a non-zero commutative ring \( R \): P: For any \( r_1, r_2 \in R \), there exists a unique \( r \in R \) such that \( r - r_1 \in J_1 \) and \( r - r_2 \in J_2 \). Q: \( J_1 + J_2 = R \) Then, which of the following statements is TRUE?
Let \( \mathbb{R}[X] \) denote the ring of polynomials in \( X \) with real coefficients. Then, the quotient ring \( \mathbb{R}[X]/(X^4 + 4) \) is
The number of non-isomorphic abelian groups of order $2^2 \cdot 3^3 \cdot 5^4$ is __________.
The number of subgroups of a cyclic group of order 12 is __________.
Let \( M \) be a \( 3 \times 3 \) real matrix such that \( M^2 = 2M + 3I \). If the determinant of \( M \) is \( -9 \), then the trace of \( M \) equals __________.
Consider \( \mathbb{R}^3 \) as a vector space with the usual operations of vector addition and scalar multiplication. Let \( x \in \mathbb{R}^3 \) be denoted by \( x = (x_1, x_2, x_3) \). Define subspaces \( W_1 \) and \( W_2 \) by \[ W_1 := \{ x \in \mathbb{R}^3 : x_1 + 2x_2 - x_3 = 0 \} \] \[ W_2 := \{ x \in \mathbb{R}^3 : 2x_1 + 3x_3 = 0 \}. \] Let \( \text{dim}(U) \) denote the dimension of the subspace \( U \). Which of the following statements are TRUE?
Let \( K \) denote the subset of \( \mathbb{C} \) consisting of elements algebraic over \( \mathbb{Q} \). Then, which of the following statements are TRUE?
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?
No. of group homomorphism from Z4 to S4