Linear Algebra
Practice QuestionsMarks 1
1
Let $A$ and $B$ be real symmetric matrices of same size. Which one of the following options is correct?
GATE ME 2025
2
Consider the system of linear equations
x + 2y + z = 5
2x + ay + 4z = 12
2x + 4y + 6z = b
The values of a and b such that there exists a non-trivial null space and the system admits infinite solutions are
GATE ME 2024
3
A linear transformation maps a point (𝑥, 𝑦) in the plane to the point (𝑥̂, 𝑦̂) according to the rule
𝑥̂ = 3𝑦, 𝑦̂ = 2𝑥.
Then, the disc 𝑥2 + 𝑦2 ≤ 1 gets transformed to a region with an area equal to _________ . (Rounded off to two decimals)
Use π = 3.14.
GATE ME 2023
4
If A = $\begin{bmatrix} 10 & 2k + 5 \\\ 3k - 3 & k + 5 \end{bmatrix} $ is a symmetric matrix, the value of k is _______.
GATE ME 2022 Set 1
5
The determinant of a $$2 \times 2$$ matrix is $$50.$$ If one eigenvalue of the matrix is $$10,$$ the other eigenvalue is __________.
GATE ME 2017 Set 2
6
The product of eigenvalues of the matrix $$P$$ is $$P = \left[ {\matrix{
2 & 0 & 1 \cr
4 & { - 3} & 3 \cr
0 & 2 & { - 1} \cr
} } \right]$$
GATE ME 2017 Set 1
7
The solution to the system of equations is $$\left[ {\matrix{
2 & 5 \cr
{ - 4} & 3 \cr
} } \right]\left\{ {\matrix{
x \cr
y \cr
} } \right\} = \left\{ {\matrix{
2 \cr
{ - 30} \cr
} } \right\}$$
GATE ME 2016 Set 1
8
A real square matrix $$A$$ is called skew-symmetric if
GATE ME 2016 Set 3
9
The condition for which the eigenvalues of the matrix $$A = \left[ {\matrix{
2 & 1 \cr
1 & k \cr
} } \right]$$ are positive, is
GATE ME 2016 Set 2
10
At least one eigenvalue of a singular matrix is
GATE ME 2015 Set 2
11
If any two columns of a determinant $$P = \left| {\matrix{
4 & 7 & 8 \cr
3 & 1 & 5 \cr
9 & 6 & 2 \cr
} } \right|$$ are interchanged, which one of the following statements regarding the value of the determinant is CORRECT?
GATE ME 2015 Set 1
12
The lowest eigen value of the $$2 \times 2$$ matrix $$\left[ {\matrix{
4 & 2 \cr
1 & 3 \cr
} } \right]$$ is ______.
GATE ME 2015 Set 3
13
Given that the determinant of the matrix $$\left[ {\matrix{
1 & 3 & 0 \cr
2 & 6 & 4 \cr
{ - 1} & 0 & 2 \cr
} } \right]$$ is $$-12$$, the determinant of the matrix $$\left[ {\matrix{
2 & 6 & 0 \cr
4 & {12} & 8 \cr
{ - 2} & 0 & 4 \cr
} } \right]$$ is
GATE ME 2014 Set 1
14
Which one of the following describes the relationship among the three vectors, $$\widehat i + \widehat j + \widehat k,\,\,2\widehat i + 3\widehat j + \widehat k$$ and $$5\widehat i + 6\widehat j + 4\widehat k?$$
GATE ME 2014 Set 1
15
Which one of the following equations is a correct identity for arbitrary $$3 \times 3$$ real matrices $$P,Q$$ and $$R$$?
GATE ME 2014 Set 4
16
Consider a $$3 \times 3$$ real symmetric matrix $$S$$ such that two of its eigen values are $$a \ne 0,$$ $$b\,\, \ne 0$$ with respective eigen vectors $$\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
{{x_3}} \cr
} } \right],\left[ {\matrix{
{{y_1}} \cr
{{y_2}} \cr
{{y_3}} \cr
} } \right].$$ If $$a\, \ne b$$ then $${x_1}{y_1} + {x_2}{y_2} + {x_3}{y_3}$$ equals
GATE ME 2014 Set 3
17
The eigen values of a symmetric matrix are all
GATE ME 2013
18
For the matrix $$A = \left[ {\matrix{
5 & 3 \cr
1 & 3 \cr
} } \right],$$ ONE of the normalized eigen vectors is given as
GATE ME 2012
19
Eigen values of a real symmetric matrix are always
GATE ME 2011
20
Consider the following system of equations
$$2{x_1} + {x_2} + {x_3} = 0,\,\,{x_2} - {x_3} = 0$$ and $${x_1} + {x_2} = 0.$$
This system has
$$2{x_1} + {x_2} + {x_3} = 0,\,\,{x_2} - {x_3} = 0$$ and $${x_1} + {x_2} = 0.$$
This system has
GATE ME 2011
21
One of the eigen vector of the matrix $$A = \left[ {\matrix{
2 & 2 \cr
1 & 3 \cr
} } \right]$$ is
GATE ME 2010
22
For a matrix $$\left[ M \right] = \left[ {\matrix{
{{3 \over 5}} & {{4 \over 5}} \cr
x & {{3 \over 5}} \cr
} } \right].$$ The transpose of the matrix is equal to the inverse of the matrix, $${\left[ M \right]^T} = {\left[ M \right]^{ - 1}}.$$ The value of $$x$$ is given by
GATE ME 2009
23
The matrix $$\left[ {\matrix{
1 & 2 & 4 \cr
3 & 0 & 6 \cr
1 & 1 & P \cr
} } \right]$$ has one eigen value to $$3.$$ The sum of the other two eigen values is
GATE ME 2008
24
For what values of 'a' if any will the following system of equations in $$x, y$$ and $$z$$ have a solution?
$$$2x+3y=4,$$$
$$$x+y+z=4,$$$
$$$x+2y-z=a$$$
GATE ME 2008
25
The number of linearly independent eigen vectors of $$\left[ {\matrix{
2 & 1 \cr
0 & 2 \cr
} } \right]$$ is
GATE ME 2007
26
If a square matrix $$A$$ is real and symmetric then the eigen values
GATE ME 2007
27
$$A$$ is a $$3 \times 4$$ matrix and $$AX=B$$ is an inconsistent system of equations. The highest possible rank of $$A$$ is
GATE ME 2005
28
For what value of $$x$$ will the matrix given below become singular? $$\left[ {\matrix{
8 & x & 0 \cr
4 & 0 & 2 \cr
{12} & 6 & 0 \cr
} } \right]$$
GATE ME 2004
29
The sum of the eigen values of the matrix $$\left[ {\matrix{
1 & 1 & 3 \cr
1 & 5 & 1 \cr
3 & 1 & 1 \cr
} } \right]$$ is
GATE ME 2004
30
For the following set of simultaneous equations
$$$1.5x - 0.5y + z = 2$$$
$$$4x + 2y + 3z = 9$$$
$$$7x + y + 5z = 10$$$
GATE ME 1997
31
In the Gauss - elimination for a solving system of linear algebraic equations, triangularization leads to
GATE ME 1996
32
The eigen values of $$\left[ {\matrix{
1 & 1 & 1 \cr
1 & 1 & 1 \cr
1 & 1 & 1 \cr
} } \right]$$ are
GATE ME 1996
33
Solve the system $$2x+3y+z=9,$$ $$4x+y=7,$$ $$x-3y-7z=6$$
GATE ME 1995
34
Among the following, the pair of vectors orthogonal to each other is
GATE ME 1995
35
Find out the eigen value of the matrix $$A = \left[ {\matrix{
1 & 0 & 0 \cr
2 & 3 & 1 \cr
0 & 2 & 4 \cr
} } \right]$$ for any one of the eigen values, find out the corresponding eigen vector?
GATE ME 1994
Marks 2
1
The matrix $\begin{bmatrix} 1 & a \\ 8 & 3 \end{bmatrix}$ (where $a > 0$) has a negative eigenvalue if $a$ is greater than
GATE ME 2024
2
A is a 3 × 5 real matrix of rank 2. For the set of homogeneous equations Ax = 0, where 0 is a zero vector and x is a vector of unknown variables, which of the following is/are true?
GATE ME 2022 Set 2
3
If the sum and product of eigenvalues of a 2 × 2 real matrix $\begin{bmatrix}3&p\\\ p&q\end{bmatrix} $ are 4 and -1 respectively, then |p| is _______ (in integer).
GATE ME 2022 Set 2
4
The system of linear equations in real (x, y) given by
$\rm \begin{pmatrix} \rm x & \rm y \end{pmatrix} \begin{bmatrix} 2 & 5- 2 α \\\ α & 1 \end{bmatrix} = \rm \begin{pmatrix} \rm 0 & \rm 0 \end{pmatrix} $
involves a real parameter α and has infinitely many non-trivial solutions for special value(s) of α. Which one or more among the following options is/are non-trivial solution(s) of (x, y) for such special value(s) of α ?
GATE ME 2022 Set 1
5
Consider the matrix $$A = \left[ {\matrix{
{50} & {70} \cr
{70} & {80} \cr
} } \right]$$ whose eigenvectors corresponding to eigen values $${\lambda _1}$$ and $${\lambda _2}$$ are $${x_1} = \left[ {\matrix{
{70} \cr
{{\lambda _1} - 50} \cr
} } \right]$$ and $${x_2} = \left[ {\matrix{
{{\lambda _2} - 80} \cr
{70} \cr
} } \right],$$ respectively. The value of $$x_1^T{x_2}$$ is ________
GATE ME 2017 Set 2
6
The number of linear independent eigenvectors of matrix $$A = \left[ {\matrix{
2 & 1 & 0 \cr
0 & 2 & 0 \cr
0 & 0 & 3 \cr
} } \right]$$ is ________.
GATE ME 2016 Set 3
7
For a given matrix $$P = \left[ {\matrix{
{4 + 3i} & { - i} \cr
i & {4 - 3i} \cr
} } \right],$$ where $$i = \sqrt { - 1} ,$$ the inverse of matrix $$P$$ is
GATE ME 2015 Set 3
8
Choose the CORRECT set of functions, which are linearly dependent.
GATE ME 2013
9
$$x+2y+z=4, 2x+y+2z=5, x-y+z=1$$
The system of algebraic equations given above has
The system of algebraic equations given above has
GATE ME 2012
10
The eigen vectors of the matrix $$\left[ {\matrix{
1 & 2 \cr
0 & 2 \cr
} } \right]$$ are written in the form $$\left[ {\matrix{
1 \cr
a \cr
} } \right]\,\,\& \,\,\left[ {\matrix{
1 \cr
b \cr
} } \right].$$ What is $$a+b$$?
GATE ME 2008
11
Eigen values of a matrix $$S = \left[ {\matrix{
3 & 2 \cr
2 & 3 \cr
} } \right]$$ are $$5$$ and $$1.$$ What are the eigen values of the matrix $${S^2} = SS?$$
GATE ME 2006
12
Multiplication of matrices $$E$$ and $$F$$ is $$G.$$ Matrices $$E$$ and $$G$$ are
$$E = \left[ {\matrix{ {\cos \theta } & { - sin\theta } & 0 \cr {sin\theta } & {\cos \theta } & 0 \cr 0 & 0 & 1 \cr } } \right]$$ and $$G = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$
What is the matrix $$F?$$
$$E = \left[ {\matrix{ {\cos \theta } & { - sin\theta } & 0 \cr {sin\theta } & {\cos \theta } & 0 \cr 0 & 0 & 1 \cr } } \right]$$ and $$G = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$
What is the matrix $$F?$$
GATE ME 2006
13
Which one of the following is an eigen vector of the matrix $$\left[ {\matrix{
5 & 0 & 0 & 0 \cr
0 & 5 & 0 & 0 \cr
0 & 0 & 2 & 1 \cr
0 & 0 & 3 & 1 \cr
} } \right]$$ is
GATE ME 2005