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

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

Most answered questions in Linear Algebra

9 votes

1 answer

361

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 17
Let $A$ be an $m \times n$-matrix and let $B$ be an $n \times m$-matrix. Then which of the following statement is not true for all such matrices? $B A$ is defined the columns of $A B$ are linear combinations of the columns of $B$ $A B$ is defined the columns of $A B$ are linear combinations of the columns of $A$

Let $A$ be an $m \times n$-matrix and let $B$ be an $n \times m$-matrix. Then which of the following statement is not true for all such matrices?$B A$ is definedthe colum...

GO Classes

790 views

GO Classes asked Mar 29, 2023

6 votes

1 answer

362

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 18
Consider $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 0\end{array}\right]$. ... not of the columns of $B$. $\boldsymbol{v}$ is neither a linear combination of the columns of $A$ nor of the columns of $B$.

Consider $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 0\end{array}\right]$. Let ...

GO Classes

571 views

GO Classes asked Mar 29, 2023

8 votes

1 answer

363

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 19
Let $A$ be a $2 \times 3$-matrix and $b$ a vector in $\mathbb{R}^2$. Consider the following two statements: $(\text{P}1)\; A$ has at most two pivots, $(\text{P2})$ Assuming $A x=b$ has a solution, then it ... $\text{P1}$ is correct. Statement $\text{P1}$ and Statement $\text{P2}$ are correct.

Let $A$ be a $2 \times 3$-matrix and $b$ a vector in $\mathbb{R}^2$.Consider the following two statements:$(\text{P}1)\; A$ has at most two pivots,$(\text{P2})$ Assuming ...

GO Classes

542 views

GO Classes asked Mar 29, 2023

23 votes

1 answer

364

GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 27
Consider the following set of (column) vectors: $X=\left\{\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right] \in \mathbb{R}^3 \mid 2 x_1+3 x_2-x_3=0\right\}.$ Which of the following statements are true? Every ... the correct option. Only I is true. Only II is true. Both I and II are true. Neither I, nor II are true.

Consider the following set of (column) vectors:$$X=\left\{\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right] \in \mathbb{R}^3 \mid 2 x_1+3 x_2-x_3=0\right\}.$$Whic...

GO Classes

667 views

GO Classes asked Mar 29, 2023

4 votes

1 answer

365

TIFR CSE 2023 | Part A | Question: 3
$A$ is an $n \times n$ matrix with real-valued entries. Further, there exists a vector $x \neq 0$ such that $A x=0$. Now consider a given vector $b$ in $\mathbb{R}^{n}$. How many possible vectors $z$ exist, so that $A z=b?$ $0$ $1$ $n-1$ $n$ Either $0$ or infinite

$A$ is an $n \times n$ matrix with real-valued entries. Further, there exists a vector $x \neq 0$ such that $A x=0$. Now consider a given vector $b$ in $\mathbb{R}^{n}$. ...

admin

449 views

admin asked Mar 14, 2023

3 votes

1 answer

366

TIFR CSE 2023 | Part A | Question: 5
Suppose $A$ is a $2 \times 2$ matrix such that the sum of the principal diagonal entries of $A$ is $10$ and the sum of the principal diagonal entries of $A^{2}$ is $20$. (For any $2 \times 2$ matrix $B$, the ... $80$ Nonzero, but cannot be uniquely determined from the above data. Cannot be uniquely determined from the above data, and could also be zero.

Suppose $A$ is a $2 \times 2$ matrix such that the sum of the principal diagonal entries of $A$ is $10$ and the sum of the principal diagonal entries of $A^{2}$ is $20$. ...

admin

329 views

admin asked Mar 14, 2023

4 votes

1 answer

367

TIFR CSE 2023 | Part A | Question: 15
Consider the $n \times n$ matrix $M$ ... $S$ have? $1$ $\left(\begin{array}{l}n \\ 2\end{array}\right)$ $n$ ! $(n !)^{2}$ $n$

Consider the $n \times n$ matrix $M$ defined as follows:$$M=\left(\begin{array}{cccc}1 & 2 & \ldots & n \\n+1 & n+2 & \ldots & 2 n \\2 n+1 & 2 n+2 & \ldots & 3 n \\\vdots...

admin

535 views

admin asked Mar 14, 2023

0 votes

1 answer

368

#self_doubt
if the determinant of matrix A is d then the determinant of the cofactor matrix of A will be d^2, is this rule correct? I am not able to satisfy this rule with a 2*2 matrix.

if the determinant of matrix A is d then the determinant of the cofactor matrix of A will be d^2, is this rule correct?I am not able to satisfy this rule with a 2*2 matri...

Dknights

297 views

Dknights asked Jan 12, 2023

0 votes

1 answer

369

ZEAL TEST
Isn’t the “No of independent rows”= RAnk of matrix?

Isn’t the “No of independent rows”= RAnk of matrix?

DAWID15

632 views

DAWID15 asked Jan 8, 2023

1 votes

1 answer

370

DRDO CSE 2022 Paper 1 | Question: 1
Compute $\left[M M^{T}\right]^{-1}$ for an orthogonal matrix where \[M=\left[\begin{array}{lll} \frac{1}{\sqrt{2}} & \frac{2}{\sqrt{2}} & \frac{-2}{\sqrt{2}} \\ \frac{-2}{\sqrt{2}} & \frac{2}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{2}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{2}{\sqrt{2}} \end{array}\right] .\]

Compute $\left[M M^{T}\right]^{-1}$ for an orthogonal matrix where\[M=\left[\begin{array}{lll}\frac{1}{\sqrt{2}} & \frac{2}{\sqrt{2}} & \frac{-2}{\sqrt{2}} \\\frac{-2}{\s...

admin

449 views

admin asked Dec 15, 2022

1 votes

1 answer

371

DRDO CSE 2022 Paper 1 | Question: 3
With no unique solution, solve for $n$ with the following system of equations $\begin{array}{r} a+b+2 c=3 \\ a+2 b+3 c=4 \\ a+4 b+n c=6 \end{array}$

With no unique solution, solve for $n$ with the following system of equations$$\begin{array}{r}a+b+2 c=3 \\a+2 b+3 c=4 \\a+4 b+n c=6\end{array}$$

admin

277 views

admin asked Dec 15, 2022

0 votes

1 answer

372

MADEEASY TESTSERIES
To justify the OPTION B they gave an example of 2*2 matrix. However we can see that row 2 is linearly dependent on row1. Even though the 2nd row looks non-zero it can be made into zero. SO am I wrong or the explanation is wrong?

To justify the OPTION B they gave an example of 2*2 matrix. However we can see that row 2 is linearly dependent on row1. Even though the 2nd row looks non-zero it can be ...

DAWID15

673 views

DAWID15 asked Dec 9, 2022

1 votes

1 answer

373

TIFR CSE 2022 | Part A | Question: 4
Consider the polynomial $p(x)=x^3-x^2+x-1$. How many symmetric matrices with integer entries are there whose characteristic polynomial is $p$? (Recall that the characteristic polynomial of a square matrix $A$ in the variable $x$ is defined to be the determinant of the matrix $(A-x I)$ where $I$ is the identity matrix.) $0$ $1$ $2$ $4$ Infinitely many

Consider the polynomial $p(x)=x^3-x^2+x-1$. How many symmetric matrices with integer entries are there whose characteristic polynomial is $p$? (Recall that the characteri...

admin

709 views

admin asked Sep 1, 2022

13 votes

1 answer

374

TIFR CSE 2022 | Part A | Question: 8
Let $A$ be the $(n+1) \times(n+1)$ matrix given below, where $n \geq 1$. For $i \leq n$, the $i$-th row of $A$ has every entry equal to $2i-1$ and the last row, i.e., the $(n+1)$-th row of $A$ has every entry equal to $-n^2$ ... $A$ has rank $n$ $A^2$ has rank $1$ All the eigenvalues of $A$ are distinct All the eigenvalues of $A$ are $0$ None of the above

Let $A$ be the $(n+1) \times(n+1)$ matrix given below, where $n \geq 1$. For $i \leq n$, the $i$-th row of $A$ has every entry equal to $2i-1$ and the last row, i.e., the...

admin

682 views

admin asked Sep 1, 2022

2 votes

1 answer

375

GO Classes Test Series 2023 | Linear Algebra | Test | Question: 9
Compute the determinant of $ A=\left[\begin{array}{cccc} 1 & -1 & 0 & 3\\ 2 & 5 & 2 & 6\\ 0 & 1 & 0 & 0\\ 1 & 4 & 2 & 1 \end{array}\right]. $

Compute the determinant of$$A=\left[\begin{array}{cccc}1 & -1 & 0 & 3\\2 & 5 & 2 & 6\\0 & 1 & 0 & 0\\1 & 4 & 2 & 1\end{array}\right].$$

GO Classes

381 views

GO Classes asked Aug 14, 2022

3 votes

1 answer

376

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

377

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

378

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

982 views

samarpita asked May 9, 2022

1 votes

1 answer

379

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

26 votes

1 answer

380

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

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