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Syllabus: Matrices, determinants, System of linear equations, Eigenvalues and eigenvectors, LU decomposition.

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \textbf{Year}& \textbf{2026 - 1}& \textbf{2026 - 2}& \textbf{2025 - 1}& \textbf{2025 - 2}& \textbf{2024 - 1}& \textbf{2024 - 2}& \textbf{2023}& \textbf{2022}& \textbf{2021 - 1}& \textbf{2021 - 2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}\\\hline \textbf{1 Mark Count}&2&1&1&2&1&0&2&1&0&1&0&1.1&2\\\hline \textbf{2 Marks Count}&0&1&1&1&1&1&0&2&1&1&0&0.9&2\\\hline \textbf{Total Marks}&2&3&3&4&3&2&2&5&2&3&\bf{2}&\bf{2.9}&\bf{5}\\\hline \end{array}}}$$

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Recent questions in Linear Algebra

4 4 votes
1 1 answer
416
416 views
GO Classes DPP | GATE CS | Linear Algebra | Diagonalization
True/False Question :If $A$ is a $2\times2$ complex matrix that is invertible and diagonalizable, and such that $A$ and $A^{2}$ have the same characteristic polynomial, t...
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8 8 votes
3 3 answers
464
464 views
GO Classes DPP | GATE CS | Linear Algebra | System of Linear Equations
For which vector(s) $b$ below does the system $ \begin{bmatrix}1 & 2 \\3 & 5 \\2 & 3\end{bmatrix} x = b $ have a solution?$ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} $ $ ...
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5 5 votes
1 1 answer
346
346 views
GO Classes DPP | GATE CS | Linear Algebra | Systems of linear equations, Rank & Consistency
Let $Ax = b$ be a system of linear equations where $A$ is an $m \times n$ matrix and $b$ is a $m \times 1$ column vector and $X$ is an $n \times1$ column vector of unknow...
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1 1 vote
1 1 answer
273
273 views
GO Classes DPP | GATE CS | Linear Algebra | Trace of a Matrix
Let $$S=\left \{ x \in\mathbb{R} \mid x=\operatorname{Trace}(A) \text{ for some } A \in M_{4} (\mathbb{R}) \text{ such that }A^{2}=A \right\}.$$Then which of the followi...
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5 5 votes
1 1 answer
317
317 views
GO Classes DPP | GATE CS | Linear Algebra | Rank of a matrix
Let $A$ be an $n \times n$ matrix with rank $k$. Consider the following statements:If $A$ has real entries, then $AA^{t}$ necessarily has rank $k$If $A$ has complex entri...
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6 6 votes
2 2 answers
273
273 views
GO Classes DPP | GATE CS | Linear Algebra | Properties of Inverse
Let $A,B,C,D$ be $n\times n$ matrices, each with non-zero determinant. If $ABCD=1$, then $B^{-1}$ is:$D^{-1}C^{-1}A^{-1}$ $CDA$ $ADC$ Does not necessarily exist.
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6 6 votes
3 3 answers
355
355 views
GO Classes DPP | GATE CS | Linear Algebra | Properties of Inverse
Let $P$, $Q$, and $R$ be $n \times n$ matrices, each with non-zero determinant. If $PQR = I,$ then $Q^{-1}$ is$RP$ $R^{-1}P^{-1}$ $PR$ $P^{-1}R^{-1}$
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6 6 votes
4 4 answers
288
288 views
GO Classes DPP | GATE CS | Linear Algebra | Matrix Multiplication
Consider two matrices $M_1$ and $M_2$ with $M_1M_2=0$ and $M_1$ is non singular. Then which of the following is true?$M_2$ is non singular $M_2$ is null matrix $M_2$ is t...
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7 7 votes
1 1 answer
274
274 views
GO Classes DPP | GATE CS | Linear Algebra | Symmetric Matrix
True/False Question :Let $A,B$ are $3 \times 3$ real matrices. Then$$det\left ( AB -BA \right )=\frac{tr\left [ \left ( AB -BA \right )^{3} \right ]}{3}.$$(Note : Mark $\...
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1 1 vote
1 1 answer
264
264 views
GO Classes DPP | GATE CS | Linear Algebra | Diagonalization
Suppose $A$ is any square matrix of size $n \times n$ with complex entries. Form the larger matrix $\begin{bmatrix}A & A \\0 & A\end{bmatrix}.$For which matrices $A$ is t...
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2 2 votes
1 1 answer
240
240 views
GO Classes DPP | GATE CS | Linear Algebra | Symmetric Matrix
True/False Question :There exist an integer $r\geq 1$ and a symmetric matrix $A \in M_{r}\left ( \mathbb{R} \right )$ such that for all $n \in \mathbb{N}$, we have: $$2^{...
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4 4 votes
3 3 answers
263
263 views
GO Classes DPP | GATE CS | Linear Algebra | Matrix Arithmetics
Consider matrices $M_1,M_2,M_3,M_4\in M_n(\mathbb{R})$ such that $M_1$ and $M_4$ are non singular and$M_1M_2M_3M_4=0$.Then which of the following is necessarily true?$M_2...
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5 5 votes
3 3 answers
301
301 views
GO Classes DPP | GATE CS | Linear Algebra | Eigen Values
The eigenvalues of the matrix $A=\begin{pmatrix}2 & 1 & 0\\0 & 2 & 1\\0 & 0 & -3\end{pmatrix}$ are$2,2,-3$ $2,-3,-3$ $1,2,-3$ $2,3,-3$
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4 4 votes
2 2 answers
241
241 views
GO Classes DPP | GATE CS | Linear Algebra | Commutativity of Matrix Multiplication
The matrices $R=\begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix}$ and $S=\begin{pmatrix} a & c\\ c & b \end{pmatrix}$ commute under multip...
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5 5 votes
2 2 answers
270
270 views
GO Classes DPP | GATE CS | Linear Algebra | Diagonalization & Eigen Vectors
Let $n\geq 2$. Which of the following statements is true for every $n\times n$ real matrix $A$ of rank one?There exist matrices $P,Q \in M_{n}\left ( \mathbb{R} \right )...
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7 7 votes
3 3 answers
269
269 views
GO Classes DPP | GATE CS | Linear Algebra | Diagonalization & Invertibility
Which of the following statements is correct for every linear transformation $T:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$ such that $T^{3}-T^{2}-T+I=0$?$T$ is invertible ...
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8 8 votes
3 3 answers
372
372 views
GO Classes DPP | GATE CS | Linear Algebra | Eigen Values
Two eigenvalues of a $3\times 3$ real matrix $Q$ are $(1+\sqrt{-1})$ and $4$. The determinant of $Q$ is ?
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6 6 votes
3 3 answers
364
364 views
GO Classes DPP | GATE CS | Linear Algebra | Properties of Determinant
Let $A$ and $B$ be invertible square matrices of order $n$, and let $k$ be a non-zero scalar. Which of the following statements is not always true ?$\det(AB^{-1}A^T)=\dfr...
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5 5 votes
3 3 answers
351
351 views
GO Classes DPP | GATE CS | Linear Algebra | Properties of Determinant
$N$ is a square matrix of order $n$ and its determinant is $7$. If each element of $N$ is multiplied by $2$, the determinant becomes $112$. The value of $n$ is $?$
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3 3 votes
2 2 answers
275
275 views
GO Classes DPP | GATE CS | Linear Algebra | Matrix Multiplication
If product of matrix $A=\begin{bmatrix}\cos^{2}\theta &\cos \theta \sin \theta \\ \cos \theta \sin \theta &\sin ^{2} \theta \end{bmatrix}$ and $B=\begin{bmatrix}\cos^{2}...
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