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

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}& \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{2020}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} & 1 &0&1&0&1&1&1&1&1&2&0&0.9&2
\\\hline\textbf{2 Marks Count} & 2 &1&1&1&1&1&2&1&0&0&0&1&2
\\\hline\textbf{Total Marks} & 5 &2&3&2&3&3&5&3&1&2&\bf{1}&\bf{2.9}&\bf{5}\\\hline
\end{array}}}$$

Recent questions in Linear Algebra

0 votes

0 answers

21

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 eigenvectors corresponding to 1 and 0 are \(\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\) and \(\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}\) respectively, then the value of \(3f\) is equal to ________________________

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 eigenvectors corresponding to 1 and 0 are...

163 views

rajveer43 asked Jan 11

0 votes

1 answer

22

GATE 2016 | MATHS | Q-11
Let \( \mathbf{v}, \mathbf{w}, \mathbf{u} \) be a basis of \( \mathbb{V} \). Consider the following statements P and Q: (P) : \( \{\mathbf{v} + \mathbf{w}, \mathbf{w} + \mathbf{u}, \mathbf{v} - \mathbf{u}\} \) is a basis of \( \mathbb{V} \). ( ... a basis of \( \mathbb{V} \). Which of the above statements hold TRUE? (A) both P and Q (B) only P (C) only Q (D) Neither P nor Q

Let \( \mathbf{v}, \mathbf{w}, \mathbf{u} \) be a basis of \( \mathbb{V} \). Consider the following statements P and Q:(P) : \( \{\mathbf{v} + \mathbf{w}, \mathbf{w} + \m...

150 views

rajveer43 asked Jan 11

0 votes

0 answers

23

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) : Let S be a diagonalizable matrix. If T is a matrix such that \( ... ), then T is diagonalizable. Which of the above statements hold TRUE? (A) both P and Q (B) only P (C) only Q (D) Neither P nor Q

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) : Let S be a diagon...

94 views

rajveer43 asked Jan 11

0 votes

0 answers

24

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 \( \text{dim}(T) < n \) and $T^2 = \lambda T$, for some \( \lambda \in \mathbb{R} \backslash \{0\} \), then which ... 0 \) for all \( X \in \mathbb{W} \) (C) \( T \) is invertible (D) \( \lambda\) is the only eigenvalue of \( T \)

Consider a real vector space \( V \) of dimension \( n \) and a non-zero linear transformation \( T: \mathbb{V} \rightarrow \mathbb{V} \). If \( \text{dim}(T) < n \) and ...

82 views

rajveer43 asked Jan 11

0 votes

0 answers

25

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 by \[ \begin{bmatrix} e^t - e^{2t} \\ -e^{t} + e^{2t} \\ e^t + e^{2t} \end{bmatrix}, \begin{bmatrix} ... \\ e^{-t} - 2e^t \\ -e^{-t} + 2e^t \end{bmatrix}, \] then the sum of the diagonal entries of \( A \) is equal to

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 by\[\begin{bm...

94 views

rajveer43 asked Jan 11

0 votes

2 answers

26

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 matrix and \( u^T \) is the transpose of \( u \). If \( \lambda \) and \( \mu \) are two distinct eigenvalues of \( A \), then \[ | \lambda - \mu | = \] _________

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 matrix and \( u^T \) is t...

102 views

rajveer43 asked Jan 11

1 votes

1 answer

27

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

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 \( \mathbb{R}^3 \), ...

146 views

rajveer43 asked Jan 11

0 votes

0 answers

28

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 \( S : M_2(\mathbb{R}) \to M_2(\mathbb{R}) \) by \( S(X) = 2X + X^T \), where \( X^T \) denotes the transpose of the matrix \( X \). Then the trace of \( S \) equals________

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 \( S : M_2(\mathbb{R...

64 views

rajveer43 asked Jan 11

0 votes

1 answer

29

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 with each diagonal entry 1 and \(U\) is upper triangular. \(Q\): The matrix \( ... \(Q\) are TRUE (C) \(P\) is FALSE and \(Q\) is TRUE (D) Both \(P\) and \(Q\) are FALSE

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 trian...

195 views

rajveer43 asked Jan 11

0 votes

0 answers

30

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_________

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 \( \math...

49 views

rajveer43 asked Jan 11

0 votes

0 answers

31

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 ._______

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...

59 views

rajveer43 asked Jan 11

0 votes

0 answers

32

Diagonalization of Matrix

Debargha Mitra Roy

61 views

Debargha Mitra Roy asked Jan 11

0 votes

0 answers

33

Diagonalization of Matrix - Orthogonal Transformation
Consider a symmetric matrix $M=\begin{bmatrix} \frac{1}{3} & 0 & \frac{2}{3}\\ 0&1 &0 \\ \frac{2}{3}&0 & \frac{1}{3} \end{bmatrix}$. An orthogonal matrix $O$ which can diagonalize this matrix by an orthogonal transformation $O^TMO$ is given by $O = $ ______

Consider a symmetric matrix $M=\begin{bmatrix} \frac{1}{3} & 0 & \frac{2}{3}\\ 0&1 &0 \\ \frac{2}{3}&0 & \frac{1}{3} \end{bmatrix}$. An orthogonal matrix $O$ which can di...

Debargha Mitra Roy

103 views

Debargha Mitra Roy asked Jan 11

0 votes

0 answers

34

GATE 2019 | Maths | DA Sample questions
Let $V$ be the vector space of all $3 \times 3$ matrices with complex entries over the real field. If $W_1 = \{A \in V : A = \bar{\mathbf{A}}^T \}$ and $W_2 = \{A \in V : trace(A)=0\}$, then the dimension of $W_1 + W_2$ is equal to ______________ ($\bar{\mathbf{A}}^T $ denotes the conjugate transpose of $A$.)

Let $V$ be the vector space of all $3 \times 3$ matrices with complex entries over the real field. If $W_1 = \{A \in V : A = \bar{\mathbf{A}}^T\}$ and $W_2 = \{A \in V : ...

112 views

rajveer43 asked Jan 10

2 votes

1 answer

35

GATE 2019 | MATHS | LINEAR ALGEBRA
Let $ \mathbf{M} $ be a $3 \times 3$ real symmetric matrix with eigenvalues $0, 2$ and $a$ with the respective eigenvectors $\mathbf{u} = \begin{bmatrix} 4 \\ b \\ c \end{bmatrix}$, $\mathbf{v} = \begin{bmatrix} -1 \\ 2 \\ 0 \end{bmatrix}$, and ... of the above statements are TRUE? (A) I, II and III only (B) I and II only (C) II and IV only (D) III and IV only

Let $ \mathbf{M} $ be a $3 \times 3$ real symmetric matrix with eigenvalues $0, 2$ and $a$ with the respective eigenvectors $\mathbf{u} = \begin{bmatrix} 4 \\ b \\ c \end...

102 views

rajveer43 asked Jan 10

1 votes

0 answers

36

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 $(\lambda - 1)(\lambda + 1)(\lambda - 2)$ respectively, then the rank of the matrix $ \mathbf{A} + \mathbf{I} $, where $ \mathbf{I} $ is the identity matrix of the appropriate order, is________________

If the characteristic polynomial and minimal polynomial of a square matrix $ \mathbf{A} $ are $(\lambda - 1)(\lambda + 1)^4 (\lambda - 2)^5$ and $(\lambda - 1)(\lambda + ...

78 views

rajveer43 asked Jan 10

0 votes

0 answers

37

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 $, where $ \det(\mathbf{M}) $ denotes the determinant of a square matrix $ \mathbf{M} $ ... $ 0 $ of $ \mathbf{A} $ is __________

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 $, where $ \det(\mathbf{M}) $ denote...

63 views

rajveer43 asked Jan 10

0 votes

0 answers

38

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 $ over $ \mathbb{R} $. Consider the following statements: $P:$ ... P and Q are TRUE (B) P is TRUE and Q is FALSE (C) P is FALSE and Q is TRUE (D) both P and Q are FALSE

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} $. Consid...

111 views

rajveer43 asked Jan 10

0 votes

0 answers

39

GATE 2021 | MATHS | Q-14
Let $ \mathbf{R} $ be the row reduced echelon form of a $ 4 \times 4 $ real matrix $ \mathbf{A} $, and let the third column of $ \mathbf{R} $ be $ \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} $. Consider the following statements: $P:$ ... : (A) both P and Q are TRUE (B) P is TRUE and Q is FALSE (C) P is FALSE and Q is TRUE (D) both P and Q are FALSE

Let $ \mathbf{R} $ be the row reduced echelon form of a $ 4 \times 4 $ real matrix $ \mathbf{A} $, and let the third column of $ \mathbf{R} $ be $ \begin{bmatrix} 0 \\ 1 ...

62 views

rajveer43 asked Jan 10

0 votes

0 answers

40

GATE 2021 | MATHS | Q-11
Let $ \mathbf{A} $ be a $ 3 \times 4 $ matrix and $ \mathbf{B} $ be a $ 4 \times 3 $ matrix with real entries such that $ \mathbf{A}\mathbf{B} $ is non-singular. Consider the following statements: $P:$ Nullity of $ \mathbf{A} $ is $ 0 $. $Q:$ ... Then (A)]both P and Q are TRUE (B) P is TRUE and Q is FALSE (C) P is FALSE and Q is TRUE (D) both P and Q are FALSE

Let $ \mathbf{A} $ be a $ 3 \times 4 $ matrix and $ \mathbf{B} $ be a $ 4 \times 3 $ matrix with real entries such that $ \mathbf{A}\mathbf{B} $ is non-singular. Consider...

61 views

rajveer43 asked Jan 10

Page:

« prev
1
2
3
4
5
6
7
...
57
next »