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

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

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Highest voted questions in Linear Algebra

1 votes
1 answer
361
GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 11
If $A = \begin{bmatrix} 1&-3 & -4\\ -1 & 3 & 4\\ 1 & -3 & -4 \end{bmatrix},$ is a nilpotent matrix, its index is _______
If $A = \begin{bmatrix}1&-3 & -4\\-1 & 3 & 4\\1 & -3 & -4\end{bmatrix},$ is a nilpotent matrix, its index is _______
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1 votes
1 answer
362
GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 13
The determinant of the matrix ... is _________
The determinant of the matrix $A = \begin{pmatrix}\frac{1}{2}& \alpha & 1 & 1 & 1 & 1 & 1 & 1\\0 & -2 & 1 & 1 & 2 & -3 & 8 & -9\\0 & 0 & -88 & 0 & 0 & 0 & 2 & 0\\0 & 0 & ...
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1 votes
1 answer
363
GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 14
The value of $\begin{vmatrix} a^{2} & a & bc\\ b^{2} & b & ca\\ c^{2} & c & ab \end{vmatrix} + \begin{vmatrix} 1 & 1 & 1\\ a^{2} & b^{2} & c^{2}\\ a^{3} & b^{3} & c^{3} \end{vmatrix} =$ ________
The value of $\begin{vmatrix}a^{2} & a & bc\\b^{2} & b & ca\\c^{2} & c & ab\end{vmatrix} + \begin{vmatrix}1 & 1 & 1\\a^{2} & b^{2} & c^{2}\\a^{3} & b^{3} & c^{3}\end{vmat...
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1 votes
1 answer
364
GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 17
If the matrix $P = \begin{bmatrix} 0 & 1 & 0 & 0\\ 0& 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \end{bmatrix}$, then the value of $\text{det(P)}$ is ______
If the matrix $P = \begin{bmatrix} 0 & 1 & 0 & 0\\0& 0 & 0 & 1\\1 & 0 & 0 & 0\\0 & 0 & 1 & 0 \end{bmatrix}$, then the value of $\text{det(P)}$ is ______
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1 votes
1 answer
366
GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 19
The system of equations given below has _____ $x + y + z + 3 = 0$ $3x + y -2z + 2 = 0$ $2x +4y + 7z -7 = 0$ a unique solution no solution infinite number of solutions exactly two solutions
The system of equations given below has _____$x + y + z + 3 = 0$$3x + y -2z + 2 = 0$$2x +4y + 7z -7 = 0$a unique solutionno solutioninfinite number of solutionsexactly tw...
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1 votes
1 answer
368
GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 24
Let $A$ be a real $2 \times 2$ matrix. If $7+5i$ is an eigenvalue of $A$, $\text{det(A)} = \alpha,$ and $\text{trace(A)} = \beta,$ then the value of $2\alpha + 3\beta$ is _______
Let $A$ be a real $2 \times 2$ matrix. If $7+5i$ is an eigenvalue of $A$, $\text{det(A)} = \alpha,$ and $\text{trace(A)} = \beta,$ then the value of $2\alpha + 3\beta$ is...
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1 votes
1 answer
369
GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 30
In the LU decomposition of the matrix $\begin{bmatrix}3 & 5\\ 6 & 8\end{bmatrix}$, if the diagonal elements of $U$ are both $1$, then the lower diagonal entry $l_{22}$ of $L$ is ________
In the LU decomposition of the matrix $\begin{bmatrix}3 & 5\\ 6 & 8\end{bmatrix}$, if the diagonal elements of $U$ are both $1$, then the lower diagonal entry $l_{22}$ of...
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1 votes
3 answers
370
NIELIT 2016 MAR Scientist C - Section B: 8
The eigenvalues of the matrix $\begin{bmatrix}1 & 2\\ 4 & 3 \end{bmatrix}$ are $\text{5 and -5}$ $\text{5 and -1}$ $\text{1 and -5}$ $\text{2 and 3}$
The eigenvalues of the matrix $\begin{bmatrix}1 & 2\\ 4 & 3 \end{bmatrix}$ are$\text{5 and -5}$$\text{5 and -1}$$\text{1 and -5}$$\text{2 and 3}$
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1 votes
1 answer
373
NIELIT 2017 DEC Scientist B - Section B: 60
Consider two matrices $M_1$ and $M_2$ with $M_1^*M_2=0$ and $M_1$ is non singular. Then which of the following is true? $M_2$ is non singular $M_2$ is null matrix $M_2$ is the identity matrix $M_2$ is transpose of $M_1$
Consider two matrices $M_1$ and $M_2$ with $M_1^*M_2=0$ and $M_1$ is non singular. Then which of the following is true?$M_2$ is non singular$M_2$ is null matrix$M_2$ is t...
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1 votes
2 answers
374
UGC NET CSE | December 2004 | Part 2 | Question: 1
$AVA=A$ is called : Identity law De Morgan’s law Idempotent law Complement law
$AVA=A$ is called :Identity lawDe Morgan’s lawIdempotent lawComplement law
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1 votes
1 answer
376
ISI2014-DCG-64
The value of $\lambda$ such that the system of equation $\begin{array}{} 2x & – & y & + & 2z & = & 2 \\ x & – & 2y & + & z & = & -4 \\ x & + & y & + & \lambda z & = & 4 \end{array}$ has no solution is $3$ $1$ $0$ $-3$
The value of $\lambda$ such that the system of equation$$\begin{array}{} 2x & – & y & + & 2z & = & 2 \\ x & – & 2y & + & z & = & -4 \\ x & + & y & + & \lambda z & = &...
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1 votes
1 answer
377
ISI2015-MMA-37
Let $a$ be a non-zero real number. Define $f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$ for $x \in \mathbb{R}$. Then, the number of distinct real roots of $f(x) =0$ is $1$ $2$ $3$ $4$
Let $a$ be a non-zero real number. Define$$f(x) = \begin{vmatrix} x & a & a & a \\ a & x & a & a \\ a & a & x & a \\ a & a & a & x \end{vmatrix}$$ for $x \in \mathbb{R}$....
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1 votes
2 answers
378
ISI2015-MMA-63
Let $\theta=2\pi/67$. Now consider the matrix $A = \begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix}$. Then the matrix $A^{2010}$ ... $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
Let $\theta=2\pi/67$. Now consider the matrix $A = \begin{pmatrix} \cos \theta & \sin \theta \\ – \sin \theta & \cos \theta \end{pmatrix}$. Then the matrix $A^{2010}$ i...
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1 votes
1 answer
380
ISI2015-DCG-11
Let two systems of linear equations be defined as follows: $\begin{array} {} & x+y & =1 \\ P: & 3x+3y & =3 \\ & 5x+5y & =5 \end{array}$ ... $P$ and $Q$ are inconsistent $P$ and $Q$ are consistent $P$ is consistent but $Q$ is inconsistent None of the above
Let two systems of linear equations be defined as follows:$\begin{array} {} & x+y & =1 \\ P: & 3x+3y & =3 \\ & 5x+5y & =5 \end{array}$ and $\begin{array} {} & x+y & =3 \\...
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