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
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Most answered questions in Linear Algebra

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1 answer

331

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...

rajveer43

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rajveer43 asked Jan 11

1 votes

1 answer

332

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 $, ...

rajveer43

155 views

rajveer43 asked Jan 11

0 votes

1 answer

333

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...

rajveer43

209 views

rajveer43 asked Jan 11

2 votes

1 answer

334

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...

rajveer43

110 views

rajveer43 asked Jan 10

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1 answer

335

GATE 2022 | MATHS | Q-65
Let M be a $3 × 3$ real matrix such that $M^2 = 2M + 3I$. If the determinant of $M$ is $−9$, then the trace of $M$ equals._______

Let M be a $3 × 3$ real matrix such that $M^2 = 2M + 3I$. If the determinant of $M$ is $−9$, then the trace of $M$ equals._______

rajveer43

161 views

rajveer43 asked Jan 10

0 votes

1 answer

336

GATE 2022 | MATHS | LINEAR ALGEBRA
Suppose that the characteristic equation of $ M \in \mathbb{C}^{3 \times 3} $ is \[ \lambda^3 + \alpha \lambda^2 + \beta \lambda - 1 = 0, \] where $ \alpha, \beta \in \mathbb{C} $ with $ \alpha + \beta \neq 0 $. Which of the following statements is TRUE? (A) $ M(I ... 1} (M^{-1} + \beta I) = M - \alpha I $ (D)$ M^{-1} (M^{-1} - \beta I) = M + \alpha I $

Suppose that the characteristic equation of $ M \in \mathbb{C}^{3 \times 3} $ is\[ \lambda^3 + \alpha \lambda^2 + \beta \lambda - 1 = 0, \]where \( \alpha, \beta \in \m...

rajveer43

132 views

rajveer43 asked Jan 10

1 votes

1 answer

337

GATE 2023 | MATHS | Question 59
Let $A$ be a $3 \times 3$ real matrix with $\text{det}(A + iI) = 0$, where $i = \sqrt{-1}$ and $I$ is the $3 \times 3$ identity matrix. If $\text{det}(A) = 3$, then the trace of $A^2$ is $\underline{\hspace{1cm}}$.

Let $A$ be a $3 \times 3$ real matrix with $\text{det}(A + iI) = 0$, where $i = \sqrt{-1}$ and $I$ is the $3 \times 3$ identity matrix. If \(\text{det}(A) = 3...

rajveer43

128 views

rajveer43 asked Jan 10

0 votes

1 answer

338

GATE 2023 | Maths | Problems for DA Paper
Consider the linear system $Mx = b$, where \[ M = \begin{bmatrix} 2 & -1 \\ -4 & 3 \end{bmatrix} \] and \[ b = \begin{bmatrix} -2 \\ 5 \end{bmatrix} \]. Suppose $M = LU$, where $L$ and $U$ are lower triangular and upper ... (A) both 𝑃 and 𝑄 are TRUE (B) 𝑃 is FALSE and 𝑄 is TRUE (C) 𝑃 is TRUE and 𝑄 is FALSE (D) both 𝑃 and 𝑄 are FALSE

Consider the linear system $Mx = b$, where \[ M = \begin{bmatrix} 2 & -1 \\ -4 & 3 \end{bmatrix} \]and\[ b = \begin{bmatrix} -2 \\ 5 \end{bmatrix} \].Suppose $M = LU$...

rajveer43

115 views

rajveer43 asked Jan 10

0 votes

1 answer

339

GATE 2023 | Maths | Linear Alegbra
Let $ M = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & 4 \\ 0 & 0 & 1 \end{bmatrix} $ and $ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $. If $ 6M^{-1} = M^2 - 6M + \alpha I $ for some $ \alpha \in \mathbb{R} $, then the value of $ \alpha $ is equal to $\underline{\hspace{1cm}}$.

Let $ M = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & 4 \\ 0 & 0 & 1 \end{bmatrix} $ and $ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $. If \( 6...

rajveer43

93 views

rajveer43 asked Jan 10

0 votes

1 answer

340

#self doubt
I found this statement in math blog.....can anyone please help on this… For a upper triangular Matrix I tried to derive with Eigen values as 1,2,0,0,0 for 5*5 but I am not getting rank as 2 for this Matrix.

I found this statement in math blog.....can anyone please help on this…For a upper triangular Matrix I tried to derive with Eigen values as 1,2,0,0,0 for 5*5 but I am n...

Dknights

116 views

Dknights asked Jan 7

0 votes

1 answer

341

#self doubt
How many skew symmetric matrices are possible with a number set = [-2,-1,1,2,3,4,0] 0 can be used at most 3 times other numbers are allowed for repetition.

How many skew symmetric matrices are possible with a number set = [-2,-1,1,2,3,4,0] 0 can be used at most 3 times other numbers are allowed for repetition.

Dknights

118 views

Dknights asked Dec 22, 2023

0 votes

1 answer

342

practice book gate
The number of linearly independent solutions of the system of equations is equal to A- 1 B- 2 C-3 D- 0

The number of linearly independent solutions of the system of equationsis equal toA- 1B- 2C-3D- 0

akshay_123

413 views

akshay_123 asked Sep 8, 2023

0 votes

1 answer

343

practice book gate
the eigen values of the matrix are (a) (a+1), 0 (b) a, 0 (c) (a-1), 0 (d) 0, 0

the eigen values of the matrix are (a) (a+1), 0(b) a, 0(c) (a-1), 0(d) 0, 0

akshay_123

290 views

akshay_123 asked Sep 7, 2023

0 votes

1 answer

344

GO class
sir in this question m should be equal to n without this how rank=n?

sir in this question m should be equal to n without this how rank=n?

JayRathi

385 views

JayRathi asked Jul 16, 2023

0 votes

1 answer

345

test series
@sachinmittal1 sir, please verify this table it is for maybe condition.

@sachinmittal1 sir, please verify this table it is for maybe condition.

JayRathi

327 views

JayRathi asked Jul 15, 2023

0 votes

1 answer

346

Possible Solutions for a System of Linear Equations
Consider a matrix of dimensions mxn and if rank(A)=n. What can be infer about the possible solutions?

Consider a matrix of dimensions mxn and if rank(A)=n. What can be infer about the possible solutions?

Sriram M

442 views

Sriram M asked Jul 3, 2023

0 votes

1 answer

347

rbr practice questions
Is the product of eigen values of a matrix equal to its determinant true for all the matrices?

Is the product of eigen values of a matrix equal to its determinant true for all the matrices?

vanshikha020

464 views

vanshikha020 asked Jul 1, 2023

1 votes

1 answer

348

matrix maths
1 -3 3 0 -5 6 0 -3 4 a 3*3 matrix is given if x,y, z are the eigan value then find xy+yz+ax? my approch if i do row transformation in c2->c2+c3 and then c2->4c2-c3, so my matrix become upper tringular matrix then ... -6 but using genral method via substract lemda from diagonal element and then determinant of matrix getting answer -3 which one is correct and why not other one

1 -3 30 -5 60 -3 4 a 3*3 matrix is given if x,y, z are the eigan value then find xy+yz+ax? my approch… if i do row transformation in c2->c2+c3and then c2-...

jugnu1337

382 views

jugnu1337 asked May 14, 2023

0 votes

1 answer

349

#self doubt

Dknights

459 views

Dknights asked Apr 18, 2023

0 votes

1 answer

350

#appliedcourse
not getting the answer by 3*3 eigen value formula – (x^3-trace(a)*x^2+sum of minors of a(x)+|a|) eigen values are given as -2,3,6

not getting the answer by 3*3 eigen value formula – (x^3-trace(a)*x^2+sum of minors of a(x)+|a|)eigen values are given as -2,3,6

Dknights

489 views

Dknights asked Apr 14, 2023

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