GATE Overflow for GATE IN - Questions without answers in Linear Algebra

GATE IN 2025 | Question: 22

Thu, 06 Mar 2025 17:18:07 +0000

If one of the eigenvectors of the matrix $A=\left[\begin{array}{cc}-1 & -1 \\ x & -4\end{array}\right]$ is along the direction of $\left[\begin{array}{c}\alpha \\ 2 \alpha\end{array}\right]$, where $\alpha$ is any non-zero real number, then the value of $x$ is _____________ (in integer).

GATE IN 2023 | Question: 1

Mon, 22 May 2023 00:15:03 +0000

Choose solution set $S$ corresponding to the systems of two equations

$$\begin{array}{r}
x-2 y+z=0 \\
x-z=0
\end{array}$$

Note: $\mathscr{R}$ denotes the set of real numbers

$S=\left\{\alpha\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right) \mid \alpha \in \mathscr{R}\right\}$
$S=\left\{\alpha\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)+\beta\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right) \mid \alpha, \beta \in \Re\right\}$
$S=\left\{\alpha\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)+\beta\left(\begin{array}{l}2 \\ 1 \\ 2\end{array}\right) \mid \alpha, \beta \in \Re\right\}$
$S=\left\{\alpha\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right) \mid \alpha \in \mathscr{R}\right\}$

GATE IN 2022 | Question: 14

Sun, 20 Mar 2022 17:33:59 +0000

Given $\text{M} = \begin{bmatrix} 2 & 3 & 7 \\ 6 & 4 & 7 \\ 4 & 6 & 14 \end{bmatrix},$ Which of the following statement(s) is/are correct?

The rank of $\text{M}$ is $2$
The rank of $\text{M}$ is $3$
The rows of $\text{M}$ are linearly independent
The determinant of $\text{M}$ is $0$

GATE IN 2022 | Question: 39

Sun, 20 Mar 2022 17:33:56 +0000

The matrix $\text{A} = \begin{bmatrix} 4 & 3 \\ 9 & -2 \end{bmatrix}$ has eigenvalues $-5$ and $7.$

The eigenvectors(s) is/are __________

$\begin{bmatrix} 1\\ 1 \end{bmatrix}$
$\begin{bmatrix} 3\\ 4 \end{bmatrix}$
$\begin{bmatrix} 2\\ -6 \end{bmatrix}$
$\begin{bmatrix} 2\\ 8 \end{bmatrix}$

GATE IN 2021 | Question: 1

Fri, 19 Feb 2021 15:48:41 +0000

Consider the row vectors $v=(1,0)$ and $w=(2,0)$. The rank of the matrix $M=2v^{T}v+3w^{T}w$, where the superscript $\text{T}$ denotes the transpose, is

GATE IN 2021 | Question: 25

Fri, 19 Feb 2021 15:48:38 +0000

The determinant of the matrix $\text{M}$ shown below is _______________.

$$M=\begin{bmatrix} 1 & 2 & 0 & 0\\ 3 & 4 & 0 & 0\\ 0 & 0 & 4 & 3\\ 0 & 0 & 2 & 1 \end{bmatrix}$$

GATE IN 2021 | Question: 38

Fri, 19 Feb 2021 15:48:36 +0000

Given $A=\begin{pmatrix} 2 & 5\\ 0 & 3 \end{pmatrix}$. The value of the determinant $\left | A^{4}-5A^{3}+6A^{2}+2I \right |=$ _______________.

GATE2020 IN: 26

Tue, 03 Nov 2020 14:26:44 +0000

Consider the matrix $M=\begin {bmatrix} 1&-1&0\\1&-2&1\\0&-1&1\end{bmatrix}$. One of the eigenvectors of $M$ is

$\begin {bmatrix} 1\\-1\\1\end{bmatrix}$
$\begin {bmatrix} 1\\1\\-1\end{bmatrix}$
$\begin {bmatrix} -1\\1\\-1\end{bmatrix}$
$\begin {bmatrix} 1\\1\\1\end{bmatrix}$

GATE2020: 3

Tue, 03 Nov 2020 04:24:37 +0000

A set of linear equations is given in the form $Ax=b$, where A is a $2\times 4$ matrix with real number entries and $b\neq 0.$ will it be possible to solve for $x$ and obtain a unique solution by multiplying both left and right sides of the equation by $A^T$ (the super script $T$ denotes the transpose) and inverting the matrix $A^TA?$ Answer is _________

Yes, it is always possible to get a unique solution for any $2\times 4$ matrix A.
No, it is not possible to get a unique solution for any $2\times 4$ matrix A.
Yes, can obtain a unique solution provided the matrix $A^T A$ is well conditioned
Yes, can obtain a unique solution provided the matrix $A$ is well conditioned

GATE IN 2017 | Question: 1

Sun, 01 Nov 2020 16:41:37 +0000

If $\text{v}$ is a non-zero vector of dimensions $3\times1$, then the matrix $A=VV^T$ has a rank = ____________.

GATE IN 2017 | Question: 2

Sun, 01 Nov 2020 16:41:37 +0000

The figure shows a shape $\text ABC$ and its mirror image $\text A_1B_1C_1$ across the horizontal axis $\text (X-axis)$. The coordinate transformation matrix that maps $\text ABC$ to $\text A_1B_1C_1$ is

$\begin{bmatrix}0&1\\1 &0\end{bmatrix}$
$\begin{bmatrix}0 &1\\-1 &0\end{bmatrix}$
$\begin{bmatrix}-1 &0\\0 &1\end{bmatrix}$
$\begin{bmatrix}1 &0\\0 &-1\end{bmatrix}$

GATE IN 2017 | Question: 4

Sun, 01 Nov 2020 16:41:37 +0000

The eigenvalues of the matrix $A=\begin{bmatrix}1 &-1 &5\\0 &5 &6\\0 &-6 &5\end{bmatrix}$ are

$-1,\;5,\;6$
$1,\;-5\pm j6$
$1,\;5\pm j6$
$1,\;5,\;5$

GATE2019 IN: 16

Sun, 10 Feb 2019 14:41:23 +0000

A 3 x 3 matrix has eigenvalues 1, 2 and 5. The determinant of the matrix is $\_\_\_\_$.

GATE2016-28

Mon, 26 Mar 2018 18:43:21 +0000

Consider the matrix $A= \begin{pmatrix} 2 & 1 & 1\\ 2& 3& 4\\ -1& -1 & -2 \end{pmatrix} $ whose eigenvalues are $1, -1$ and $3$. Then Trace of $(A^3-3A^2)$ is $\_\_\_\_\_\_.$

GATE2015-11

Mon, 26 Mar 2018 18:32:04 +0000

Let $A$ be an $n\times n$ matrix with rank $r(0<r<n).$ Then $\text{Ax=0}$ has $p$ independent solutions, where $p$ is

$r$
$n$
$n-r$
$n+r$

GATE2014-26

Sun, 25 Mar 2018 23:10:48 +0000

A scalar valued function is defined as $f(X)=X^TAX+b^TX+c,$ where A is a symmetric positive definite matrix with dimension $n\times n$; b and X are vectors of dimension $n\times 1$. The minimum value of f(x) will occur when X equals

$(A^TA)^{-1}b$
$-(A^TA)^{-1}b$
$-(\frac{A^{-1}b}{2})$
$\frac{A^{-1}b}{2}$

GATE2014-28

Sun, 25 Mar 2018 23:10:48 +0000

For the matrix $A$ satisfying the equation given below, the eigenvalues are

$$[A] \begin {bmatrix} 1&2&3\\7&8&9\\4&5&6\end{bmatrix}=\begin {bmatrix} 1&2&3\\4&5&6\\7&8&9\end{bmatrix}$$

$(1,-j,j)$
$(1,1,0)$
$(1,1,-1)$
$(1,0,0)$

GATE IN 2013 | Question: 27

Sun, 25 Mar 2018 19:30:32 +0000

One pair of eigenvectors corresponding to the two eigenvalues of the matrix $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ is

$\begin{bmatrix}1\\-j\end{bmatrix}$,$\begin{bmatrix}j\\-1\end{bmatrix}$
$\begin{bmatrix}0\\1\end{bmatrix}$,$\begin{bmatrix}-1\\0\end{bmatrix}$
$\begin{bmatrix}1\\j\end{bmatrix}$,$\begin{bmatrix}0\\1\end{bmatrix}$
$\begin{bmatrix}1\\j\end{bmatrix}$,$\begin{bmatrix}j\\1\end{bmatrix}$

GATE IN 2013 | Question: 1

Sun, 25 Mar 2018 19:30:28 +0000

The dimension of the null space of the matrix $\begin{bmatrix} 0&1&1\\1&-1&0\\-1&0&-1 \end{bmatrix}$ is

GATE IN 2012 | Question: 27

Sun, 25 Mar 2018 09:53:04 +0000

Given that

$A=\begin{bmatrix}-5 &-3\\2 &0\end{bmatrix}$ and $I=\begin{bmatrix}1 & 0\\0 &1\end{bmatrix}$, the value of $A^3$ is

$15A+12I$
$19A+30I$
$17A+15I$
$17A+21I$

GATE2018IN: 28

Tue, 20 Feb 2018 12:03:44 +0000

Consider the following system of linear equations:

3x + 2ky = -2

kx + 6y =2

Here x and y are the unknows and k is real constant. The value of k for which there are infinite number of solutions is

3
1
-3
-6

GATE2018IN: 1

Tue, 20 Feb 2018 12:03:41 +0000

Let N be a 3 by 3 matrix with real numbers entries. The matrix N is such that N$^2$ = 0. The eigen values of N are

0, 0, 0
0,0,1
0,1,1
1,1,1