GATE MA Linear Algebra — practice questions
28 free MCQs with worked solutions. Tap any question for the answer + explanation, or practice them all in the app.
Practice GATE MA Linear Algebra in the app →A VECTOR SPACE $V$ over a field $\mathbb{F}$ requires which axiom about scalar multiplication?A set of vectors $\{v_1, \ldots, v_n\}$ is LINEARLY INDEPENDENT iffA BASIS of a vector space $V$ is a set that isThe DIMENSION of $\mathbb{R}^n$ isA LINEAR TRANSFORMATION $T : V \to W$ satisfiesThe KERNEL (null space) of a linear transformation $T : V \to W$ isThe RANK-NULLITY THEOREM states that for $T : V \to W$ on a finite-dim $V$For an $m \times n$ matrix $A$, its RANK equalsThe system $Ax = b$ has a UNIQUE solution iffFor a $3 \times 3$ matrix $A$, $\det(A^2)$ equalsA vector space $V$ has dimension 5. The maximum number of LINEARLY INDEPENDENT vectors that can be chosen fromIs $\{(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)\}$ linearly INDEPENDENT in $\mathbb{R}^3$?If $A$ is a $3 \times 3$ matrix with rank 2, the dimension of $\ker A$ isWhich is a SUBSPACE of $\mathbb{R}^3$?An EIGENVECTOR $v$ of $A$ satisfiesThe CHARACTERISTIC POLYNOMIAL of an $n \times n$ matrix $A$ isThe sum of the eigenvalues of an $n \times n$ matrix $A$ equalsThe product of the eigenvalues of a square matrix $A$ equalsFind the eigenvalues of $\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$.A square matrix $A$ is DIAGONALISABLE if and only ifCAYLEY-HAMILTON THEOREM states that for any $n \times n$ matrix $A$ with characteristic polynomial $p$For a SYMMETRIC real matrix $A$, all eigenvalues areAn ORTHOGONAL matrix $Q$ satisfiesTwo vectors $u, v$ are ORTHOGONAL iffGRAM-SCHMIDT process converts a basis intoThe TRACE of $A = \begin{pmatrix} 2 & 5 \\ 7 & 3 \end{pmatrix}$ isIf $A$ is invertible, the eigenvalues of $A^{-1}$ areLet $A$ be a $3 \times 3$ matrix with eigenvalues 1, 2, 3. Then $\det(A)$ equals