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?

Let A be a $3 \times 4$ matrix and B be a $4 \times 3$ matrix with real entries such that $ AB $ is non-singular. Consider the following statements:
P: Nullity of A is 0.
Q: $ BA $ is a non-singular matrix.
Then:

Let $ A $ be a square matrix such that \[ \text{det}(xI - A) = x^4(x - 1)^2(x - 2)^3, \] where $ \text{det}(M) $ denotes the determinant of a square matrix $ M $. If \[ \text{rank}(A^2) < \text{rank}(A^3) = \text{rank}(A^4), \] then the geometric multiplicity of the eigenvalue 0 of $ A $ is $\underline{\hspace{1cm}}$ .

Let $ V = \{ p : p(x) = a_0 + a_1 x + a_2 x^2, a_0, a_1, a_2 \in \mathbb{R} \} $ be the vector space of all polynomials of degree at most 2 over the real field $ \mathbb{R} $. Let $ T: V \to V $ be the linear operator given by
\[ T(p) = (p(0) - p(1)) + (p(0) + p(1)) x + p(0) x^2. \] Then the sum of the eigenvalues of $ T $ is $\underline{\hspace{2cm}}$ .

Let $ \langle \cdot, \cdot \rangle : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} $ be an inner product on the vector space $ \mathbb{R}^n $ over $ \mathbb{R} $. Consider the following statements: P: $ |\langle u, v \rangle| \leq \frac{1}{2} \left( \langle u, u \rangle + \langle v, v \rangle \right) $ for all $ u, v \in \mathbb{R}^n $. Q: If $ \langle u, v \rangle = \langle 2u, -v \rangle $ for all $ v \in \mathbb{R}^n $, then $ u = 0 $. Then, which of the following is correct?

Let $ T: \mathbb{R}^3 \to \mathbb{R}^3 $ be a linear transformation such that \[ T \left( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}, T^2 \left( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, T^2 \left( \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} \right) = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}. \] Then the rank of $ T $ is $\underline{\hspace{2cm}}$ .

Let $ R $ be the row reduced echelon form of a $ 4 \times 4 $ real matrix $ A $ and let the third column of $ R $ be \[ \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}. \] Consider the following statements: \[ P: \text{If} \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} \text{ is a solution of } A x = 0, \text{ then } \gamma = 0. \] \[ Q: \text{For all } b \in \mathbb{R}^4, \text{rank}[A | b] = \text{rank}[R | b]. \] Then: \\

No. of group homomorphism from Z₄ to S₄