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

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2024-1} & \textbf{2024-2} & \textbf{2023} & \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} &1&0&2& 1 &0&1&0&0.83&2
\\\hline\textbf{2 Marks Count} &1&1&0& 2 &1&1&0&1&2
\\\hline\textbf{Total Marks} &3&3&2& 5 &2&3&\bf{2}&\bf{3}&\bf{5}\\\hline
\end{array}}}$$

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Questions without an upvoted answer in Linear Algebra

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non-zero determinant of a square matrix and solution of Ax=b
if $det(A) \neq 0$ for a square matrix $A$, then a unique solution always exists for the system $Ax = b$ for any vector $b$. why ?
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10
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determinant of square matrix with linearly independent columns
If a square matrix $A_{n \times n}$ has $n$ linearly independent columns, then $det(A) \neq 0$. why ?
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8
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linear combination of vectors
why linear combination of linearly independent vectors is unique ?
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64
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Go classes linear algebra lecture filling a vector space.
Why does linear combination of 2 linearly independent vectors produce every vector in R^2 ?
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Go Classes LA for GATE DA | Homework: 2 | question: 6
Q: The given matrix has solution for:$\begin{bmatrix} 1 & 1 & 3\\ 1 & 2 & 5\\ 2 & 3 & 8 \end{bmatrix}$a. All vectors b in $\mathbb{R}^{3}$ ... ?2) why option D is incorrect ?
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166
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IIT Madras MS DSAI Written Test 2024
Given, that the eigen values of a 2 x 2 matrix are -1,1 and its singular values are 1,0. What is the rank of the matrix?a) rank is 0b) rank is 1c) Such a matrix can't existd) rank is 2
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153
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1 answers
4 votes
GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 1
Let $T_{1}, T_{2}: R^{5} \rightarrow R^{3}$ be linear transformations s.t $\operatorname{rank}\left(T_{1}\right)=3$ and nullity $\left(T_{2}\right)=3$ ... s.t $T_{3}\left(T_{1}\right)=T_{2}$. Then find rank of $T_{3}$
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144
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2 votes
GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 14
Suppose $A$ is an $11 \times 5$ matrix and $T$ is the corresponding linear transformation given by the formula $T(x)=A x$. Which of the following statements are ... $\operatorname{rank}(A) \leq 4$.
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791
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GATE DS&AI 2024 | Question: 25
Consider the $3 \times 3$ matrix $\boldsymbol{M}=\left[\begin{array}{lll}1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6\end{array}\right]$.The determinant of $\left(\boldsymbol{M}^{2}+12 \boldsymbol{M}\right)$ is $\_\_\_\_\_\_\_\_\_$.
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301
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Memory Based GATE DA 2024 | Question: 5
Consider a matrix \(M \in \mathbb{R}^{3 \times 3}\) and let \(U\) be a 2-dimensional subspace such that \(M\) is a projection onto \(U\). Which of the ... M\)The nullspace of \(M\) is 1-dimensional.The nullspace of \(M\) is 2-dimensional.
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265
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Memory Based GATE DA 2024 | Question: 29
Consider the vector \( u = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{bmatrix} \), and let \( M = uu^{\top} \). If \( \sigma_1, \sigma_2, \sigma_3, \ldots ... the singular values of \( M \), what is the value of \( \sum_{i=1}^5 \sigma_i \)?
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197
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Memory Based GATE DA 2024 | Question: 60
Linear Algebra Question: Four options were given related to subspace R3.Something like this :A. \( \alpha \cdot x + \beta \cdot y \)B. \( \alpha^2 \cdot x + \beta^2 \cdot y \)C. \(f(x) = 4x_1 + 2x_3 + 3x_3 \)D.
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909
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1 answers
4 votes
GO Classes Test Series 2024 | Mock GATE | Test 11 | Question: 30
Let $A$ be a matrix defined as $A=u v^T$, where $u$ and $v$ are column vectors of dimension $3 \times 1$. The resulting matrix $A$ will be of ... $3 \times 3$. What are the maximum number of nonzero eigenvalues possible for the matrix $A?$
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206
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GATE 2016 | MATHS | QUESTION-47
Let \( A = \begin{bmatrix} a & b & c \\ b& d & e\\ c& e& f\end{bmatrix} \) be a real matrix with eigenvalues 1, 0, and 3. If the ... -1 \\ 0 \end{bmatrix}\) respectively, then the value of \(3f\) is equal to ________________________
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125
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GATE 2016 | MATHS | Q-12
Consider the following statements P and Q:(P) : If \( M = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{bmatrix} \), then M is singular.(Q) ... above statements hold TRUE?(A) both P and Q (B) only P(C) only Q (D) Neither P nor Q
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120
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GATE 2016 | MATHS | Q-14
Consider a real vector space \( V \) of dimension \( n \) and a non-zero linear transformation \( T: \mathbb{V} \rightarrow \mathbb{V} \). If \( \text{dim}(T) < n ... ) \( T \) is invertible(D) \( \lambda\) is the only eigenvalue of \( T \)
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125
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GATE 2018 | MATHS | Q-55
Let \( A \) be a \(3 \times 3\) matrix with real entries. If three solutions of the linear system of differential equations \(\dot{x}(t) = Ax(t)\) are given ... ^t\end{bmatrix},\]then the sum of the diagonal entries of \( A \) is equal to
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140
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2 answers
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GATE 2018 | MATHS | Q-52
Consider the matrix \( A = I_9 - 2u^T u \) with \( u = \frac{1}{3}[1, 1, 1, 1, 1, 1, 1, 1, 1] \), where \( I_9 \) is the \(9 \times 9\) identity ... ( \mu \) are two distinct eigenvalues of \( A \), then \[ | \lambda - \mu | = \] _________
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222
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1 answers
1 votes
GATE 2018 | MATHS | Q-51
Consider \( \mathbb{R}^3 \) with the usual inner product. If \( d \) is the distance from \( (1, 1, 1) \) to the subspace ${(1, 1, 0), (0, 1, 1)}$ of \( \mathbb{R}^3 \), then \( 3d^2 \) is given by
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GATE 2018 | MATHS | Q-50
Let \( M_2(\mathbb{R}) \) be the vector space of all \( 2 \times 2 \) real matrices over the field \( \mathbb{R} \). Define the linear transformation \( ... the transpose of the matrix \( X \). Then the trace of \( S \) equals________
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305
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GATE 2018 | MATHS | Q-42 DA Practice Questions
Consider the following two statements:\(P\): The matrix \(\begin{bmatrix} 0 & 5 \\ 0 & 7 \end{bmatrix}\) has infinitely many LU factorizations, where \(L\) is lower triangular ... is FALSE and \(Q\) is TRUE(D) Both \(P\) and \(Q\) are FALSE
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GATE 2018 | MATHS | Q-24
Consider the subspaces\[ W_1 = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1 = x_2 + 2x_3 \} \]\[ W_2 = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1 = 3x_2 + 2x_3 \} \]of \( \mathbb{R}^3 \). Then the dimension of \(W_1 + W_2\) equals_________
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GATE 2018 | MATHS | Q-23
Let A = A = \begin{bmatrix}a & 2f & 0 \\2f & b & 3f \\0 & 3f & c \\\end{bmatrix}, where $a, b, c, f$ are real numbers and $f not equalto0$. The geometric multiplicity of the largest eigenvalue of A equals ._______
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Diagonalization of Matrix - Orthogonal Transformation
Consider a symmetric matrix $M=\begin{bmatrix} \frac{1}{3} & 0 & \frac{2}{3}\\ 0&1 &0 \\ \frac{2}{3}&0 & \frac{1}{3} \end{bmatrix}$. ... $O^TMO$ is given by $O = $ ______
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145
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GATE 2019 | Maths | DA Sample questions
Let $V$ be the vector space of all $3 \times 3$ ... $\bar{\mathbf{A}}^T$ denotes the conjugate transpose of $A$.)
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115
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1 votes
GATE 2019 | MATHS | QUESTION 40
If the characteristic polynomial and minimal polynomial of a square matrix $ \mathbf{A} $ are $(\lambda - 1)(\lambda + 1)^4 (\lambda - 2)^5$ and ... $ \mathbf{I} $ is the identity matrix of the appropriate order, is________________
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GATE 2021 | MATHS | QUESTION
Let $ \mathbf{A} $ be a square matrix such that $ \det(\mathbf{xI} - \mathbf{A}) = \mathbf{x}^4 (\mathbf{x} - 1)^2 (\mathbf{x} - 2)^3 $, ... $ 0 $ of $ \mathbf{A} $ is __________
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GATE 2021 | MATHS | PRACTICE PROBLEMS FOR DA PAPER
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 $ ... TRUE and Q is FALSE(C) P is FALSE and Q is TRUE(D) both P and Q are FALSE
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