Recent questions tagged linear-algebra / GATE Overflow for GATE CSE

3 votes

1 answer

301

GO Classes Test Series 2023 | Linear Algebra | Test | Question: 15
Let $A$ be an $n \times n$ real skew-symmetric matrix The trace of a real skew-symmetric matrix is always equal to $0.$ If $A$ is skew symmetric matrix, then $A^{2}$ is a symmetric matrix. If $n$ is odd, $A$ is not invertible If $n$ is even, $A$ is invertible

Let $A$ be an $n \times n$ real skew-symmetric matrixThe trace of a real skew-symmetric matrix is always equal to $0.$If $A$ is skew symmetric matrix, then $A^{2}$ is a s...

GO Classes

480 views

GO Classes asked Aug 14, 2022

0 votes

1 answer

302

ISI 2020 | PCB Mathematics | Question: 3
Suppose $A$ is an $(n \times n)$ matrix over $\mathbb{R}$ such that $A^{p}=0$ for some positive integer $p$. Prove that $I+A$ is an invertible matrix, where $I$ is the $(n \times n)$ identity matrix. Find the characteristic polynomial of $A$.

Suppose $A$ is an $(n \times n)$ matrix over $\mathbb{R}$ such that $A^{p}=0$ for some positive integer $p$.Prove that $I+A$ is an invertible matrix, where $I$ is the $(n...

admin

481 views

admin asked Aug 8, 2022

2 votes

1 answer

303

GATE-2014 EC
Which one of the following statements is NOT true for a square matrix A? If A is real symmetric, the eigen values of A are the diagonal elements of it. If all the principal minors of A are positive, all the eigen values of A are also positive. My question is what is “principal minors of A” ?

Which one of the following statements is NOT true for a square matrix A?If A is real symmetric, the eigen values of A are the diagonal elements of it.If all the principal...

samarpita

980 views

samarpita asked May 9, 2022

1 votes

1 answer

304

Eigen Values of an orthogonal matrix will always be +1 and the modulus will also be always |1|?
Eigen Values of an orthogonal matrix will always be +1 and the modulus will also be always |1|? I have a doubt on this point .

Eigen Values of an orthogonal matrix will always be +1 and the modulus will also be always |1|? I have a doubt on this point .

samarpita

9.3k views

samarpita asked May 9, 2022

19 votes

4 answers

305

GATE CSE 2022 | Question: 10
Consider the following two statements with respect to the matrices $\textit{A}_{m \times n}, \textit{B}_{n \times m}, \textit{C}_{n \times n}$ and $ \textit{D}_{n \times n}.$ Statement $1: tr \text{(AB)} = tr \text{(BA)}$ ... $2$ is correct. Both Statement $1$ and Statement $2$ are correct. Both Statement $1$ and Statement $2$ are wrong.

Consider the following two statements with respect to the matrices $\textit{A}_{m \times n}, \textit{B}_{n \times m}, \textit{C}_{n \times n}$ and $ \textit{D}_{n \times ...

Arjun

10.9k views

Arjun asked Feb 15, 2022

22 votes

5 answers

306

GATE CSE 2022 | Question: 35
Consider solving the following system of simultaneous equations using $\text{LU}$ decomposition. $x_{1} + x_{2} - 2x_{3} = 4$ $x_{1} + 3x_{2} - x_{3} = 7$ $2x_{1} + x_{2} - 5x_{3} = 7$ where $\textit{L}$ and $\textit{U}$ ... $\textit{L}_{32}= - \frac{1}{2}, \textit{U}_{33}= - \frac{1}{2}, x_{1}= 0$

Consider solving the following system of simultaneous equations using $\text{LU}$ decomposition.$$x_{1} + x_{2} – 2x_{3} = 4$$$$x_{1} + 3x_{2} – x_{3} = 7$$$$2x_{1} +...

Arjun

11.4k views

Arjun asked Feb 15, 2022

26 votes

1 answer

307

GATE CSE 2022 | Question: 43
Which of the following is/are the eigenvector(s) for the matrix given below? $\begin{pmatrix} - 9 & - 6 & - 2 & - 4 \\ - 8 & - 6 & - 3 & - 1 \\ 20 & 15 & 8 & 5 \\ 32 & 21 & 7 & 12 \end{pmatrix}$ ... $\begin{pmatrix} - 1 \\ 0 \\ 2 \\ 2 \end{pmatrix}$ $\begin{pmatrix} 0 \\ 1 \\ - 3 \\ 0 \end{pmatrix}$

Which of the following is/are the eigenvector(s) for the matrix given below?$$\begin{pmatrix} – 9 & – 6 & – 2 & – 4 \\ – 8 & – 6 & – 3 & – ...

Arjun

10.4k views

Arjun asked Feb 15, 2022

0 votes

0 answers

308

(Solved now) Gateoverflow test doubt
In this question – https://gateoverflow.in/348646/Gate-overflow-linear-algebra-1-10 for $A=\begin{bmatrix} 0 & i\\ -i& 0 \end{bmatrix}$ how is (B) valid ?

In this question – https://gateoverflow.in/348646/Gate-overflow-linear-algebra-1-10 for$A=\begin{bmatrix} 0 & i\\ -i& 0 \end{bmatrix}$how is (B) valid ?

o

522 views

o asked Jan 31, 2022

2 votes

1 answer

309

GATE Overflow Test Series | Mock GATE | Test 6 | Question: 24
If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \\ \end{bmatrix}, $ then $A^{2027}-A^{2022} = $ $A^5-A$ $A^6$ $A^7-A^{2}$ $A^6-A$

If $A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 1 \\1 & 0 & 0 \\\end{bmatrix}, $ then $A^{2027}-A^{2022} = $$A^5-A$$A^6$$A^7-A^{2}$$A^6-A$

Arjun

248 views

Arjun asked Jan 30, 2022

2 votes

1 answer

310

GATE Overflow Test Series | Mock GATE | Test 6 | Question: 34
The trace of the following matrix is _______ $\begin{bmatrix} 3 & 3 & 3 & 3 & 3 \\ 3 & 2 & 2 & 2 & 3 \\ 3 & 2 & 1 & 2 & 3 \\ 3 & 2 & 2 & 2 & 3 \\ 3 & 3 & 3 & 3 & 3 \\ \end{bmatrix}$

The trace of the following matrix is _______$$\begin{bmatrix}3 & 3 & 3 & 3 & 3 \\3 & 2 & 2 & 2 & 3 \\3 & 2 & 1 & 2 & 3 \\3 & 2 & 2 & 2 & 3 \\3 & 3 & 3 & 3 & 3 \\\end{bmat...

Arjun

96 views

Arjun asked Jan 30, 2022

2 votes

1 answer

311

GATE Overflow Test Series | Mock GATE | Test 6 | Question: 35
What is the minimum value of $x^{T}Ax$ subject to $x^{T} x=1,$ if $A = \begin{bmatrix} 2022 &2021 \\ 2021 & 2022 \\ \end{bmatrix}.$ $-1$ $4043$ $1$ $-4043$

What is the minimum value of $x^{T}Ax$ subject to $x^{T} x=1,$ if $A = \begin{bmatrix}2022 &2021 \\2021 & 2022 \\\end{bmatrix}.$$-1$$4043$$1$$-4043$

Arjun

326 views

Arjun asked Jan 30, 2022

0 votes

1 answer

312

linear algebra - eigenvalues
What are the eigenvalues of the following matrix?

What are the eigenvalues of the following matrix?

atulcse

409 views

atulcse asked Jan 13, 2022

0 votes

0 answers

313

made easy test series - linear algebra
If 2 and -4 are the eigenvalues of a non-singular matrix A and |A| = -8, then the eigenvalues of adj(A) are x and -y. What is the value of x+y?

If 2 and -4 are the eigenvalues of a non-singular matrix A and |A| = -8, then the eigenvalues of adj(A) are x and -y. What is the value of x+y?

atulcse

417 views

atulcse asked Jan 13, 2022

3 votes

1 answer

314

#self doubt
How to solve this question?

How to solve this question?

samarpita

601 views

samarpita asked Dec 28, 2021

1 votes

1 answer

315

Self doubt: what are eigen values of resultant matrix when add two matrices?
if eigen values of matrix A = a,b,c if eigen values of matrix B = x,y,z then is it holds every time that eigen value of A+B = a+x, b+y, c+z ? please give some reference to you answer, thankyou.

if eigen values of matrix A = a,b,cif eigen values of matrix B = x,y,zthen is it holds every time that eigen value of A+B = a+x, b+y, c+z ?please give some reference to y...

jaswanth431

279 views

jaswanth431 asked Dec 12, 2021

0 votes

1 answer

316

CMI-2021-Data Science

rsansiya111

259 views

rsansiya111 asked Dec 9, 2021

2 votes

1 answer

317

CMI-2021 Data Science
Which of the following statement(s) is/are true for an arbitrary n × n matrix A? (a) Exchanging two rows of A does not change its determinant. (b) Exchanging two rows of A does not change its trace. (c) Replacing each diagonal element of A with a 1 does not change its determinant. (d) Exchanging two columns of A negates its determinant.

Which of the following statement(s) is/are true for an arbitrary n × n matrix A?(a) Exchanging two rows of A does not change its determinant.(b) Exchanging two rows of A...

rsansiya111

293 views

rsansiya111 asked Dec 9, 2021

0 votes

0 answers

318

NPTEL Assignment Question
Check whether the following set of vectors are linearly dependent or independent. (a) S = {(1, 2, −2, −1),(2, 1, −1, 4),(−3, 0, 3, −2)} (b) S = {(1, 3, −2, 5, 4),(1, 4, 1, 3, 5),(1, 4, 2, 4, 3),(2, 7, −3, 6, 13)}

Check whether the following set of vectors are linearly dependent or independent.(a) S = {(1, 2, −2, −1),(2, 1, −1, 4),(−3, 0, 3, −2)}(b) S = {(1, 3, −2, 5, 4...

rsansiya111

252 views

rsansiya111 asked Dec 6, 2021

0 votes

0 answers

319

NPTEL Assignment Question
Show that the set F = ({1, 0}, +, .) is a field where + and . are defined as 1+1=0, 0+0=0, 0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.

Show that the set F = ({1, 0}, +, .) is a field where + and . are defined as 1+1=0, 0+0=0, 0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.

rsansiya111

374 views

rsansiya111 asked Dec 6, 2021

2 votes

1 answer

320

Applied Grand Test 2
What is the value of the given determinant ?

What is the value of the given determinant ?

LRU

764 views

LRU asked Nov 22, 2021

5 votes

4 answers

321

Applied Test Series
Consider the system of equations x+y+z=3 x+2y+3z=4 x+4y+kz=6 The above system of equations will not have a unique solution for k equals to___

Consider the system of equations x+y+z=3x+2y+3z=4x+4y+kz=6 The above system of equations will not have a unique solution for k equals to___

LRU

1.0k views

LRU asked Nov 5, 2021

2 votes

1 answer

322

Linear Algebra Eigenvalues
If the eigenvalues of a 3x3 matrix A are 1,2 and 3 then inverse of A is?

If the eigenvalues of a 3x3 matrix A are 1,2 and 3 then inverse of A is?

theakatsuki4

4.7k views

theakatsuki4 asked Sep 16, 2021

2 votes

1 answer

323

TIFR CSE 2021 | Part A | Question: 3
Let $M$ be an $n \times m$ real matrix. Consider the following: Let $k_{1}$ be the smallest number such that $M$ can be factorized as $A \cdot B$, where $A$ is an $n \times k_{1}$ and $B$ is a $k_{1} \times m$ matrix. Let $k_{2}$ be the smallest number ... $k_{1}= k_{2}= k_{3}$ No general relationship exists among $k_{1}, k_{2}$ and $k_{3}$

Let $M$ be an $n \times m$ real matrix. Consider the following:Let $k_{1}$ be the smallest number such that $M$ can be factorized as $A \cdot B$, where $A$ is an $n \time...

soujanyareddy13

651 views

soujanyareddy13 asked Mar 25, 2021

3 votes

1 answer

324

TIFR CSE 2021 | Part B | Question: 8
Let $A$ and $B$ be two matrices of size $n \times n$ and with real-valued entries. Consider the following statements. If $AB = B$, then $A$ must be the identity matrix. If $A$ is an idempotent (i.e. $A^{2} = A$) nonsingular matrix, then $A$ must be the identity matrix. ... true of $A$? $1, 2 $ and $3$ Only $2$ and $3$ Only $1$ and $2$ Only $1$ and $3$ Only $2$

Let $A$ and $B$ be two matrices of size $n \times n$ and with real-valued entries. Consider the following statements.If $AB = B$, then $A$ must be the identity matrix.If ...

soujanyareddy13

694 views

soujanyareddy13 asked Mar 25, 2021

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