Linear Algebra
Practice QuestionsMarks 1
1
The relationship between two variables $x$ and $y$ is given by $x+p y+q=0$ and is shown in the figure. Find the values of $p$ and $q$.
Note: The figure shown is representative.
GATE EE 2025
2
Let $v_1$ and $v_2$ be the two eigen vectors corresponding to distinct eigen values of a $3 \times 3$ real symmetric matrix. Which one of the following statements is true?
GATE EE 2025
3
Let $A=\left[\begin{array}{ccc}1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1\end{array}\right]$ and $b=\left[\begin{array}{c}1 / 3 \\ -1 / 3 \\ 0\end{array}\right]$, then the system of linear equations $A x=b$ has
GATE EE 2025
4
Let $P=\left[\begin{array}{ccc}2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$ and let $I$ be the identity matrix. Then $P^2$ is equal to
GATE EE 2025
5
Which one of the following matrices has an inverse?
GATE EE 2024
6
The sum of the eigenvalues of the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2$ is ______ (rounded off to the nearest integer).
GATE EE 2024
7
For a given vector $${[\matrix{ 1 & 2 & 3 \cr } ]^T}$$, the vector normal to the plane defined by $${w^T}x = 1$$ is
GATE EE 2023
8
In the figure, the vectors u and v are related as : Au = v by a transformation matrix A. The correct choice of A is
GATE EE 2023
9
Consider a 3 $$\times$$ 3 matrix A whose (i, j)-th element, ai,j = (i $$-$$ j)3. Then the matrix A will be
GATE EE 2022
10
The matrix $$A = \left[ {\matrix{
{{3 \over 2}} & 0 & {{1 \over 2}} \cr
0 & { - 1} & 0 \cr
{{1 \over 2}} & 0 & {{3 \over 2}} \cr
} } \right]$$ has three distinct eigen values and one of its eigen vectors is $$\left[ {\matrix{
1 \cr
0 \cr
1 \cr
} } \right].$$ Which one of the following can be another eigen vector of $$A$$?
GATE EE 2017 Set 1
11
$$A$$ $$3 \times 3$$ matrix $$P$$ is such that , $${p^3} = P.$$ Then the eigen values of $$P$$ are
GATE EE 2016 Set 2
12
Consider $$3 \times 3$$ matrix with every element being equal to $$1.$$ Its only non-zero eigenvalue is __________.
GATE EE 2016 Set 1
13
If the sum of the diagonal elements of a $$2 \times 2$$ matrix is $$-6$$, then the maximum possible value of determinant of the matrix is ____________.
GATE EE 2015 Set 1
14
We have a set of $$3$$ linear equations in $$3$$ unknown. $$'X \equiv Y'$$ means $$X$$ and $$Y$$ are equivalent statements and $$'X \ne Y'$$ means $$X$$ and $$y$$ are not equivalent statements.
Which one of the following is TRUE?
$$P:$$ There is a unique solution.
$$Q:$$ The equations are linearly independent .
$$R:$$ All eigen values of the coefficient matrix are non zero .
$$S:$$ The determinant of the coefficient matrix is non-zero .
Which one of the following is TRUE?
GATE EE 2015 Set 2
15
Which one of the following statements is true for all real symmetric matrices?
GATE EE 2014 Set 2
16
Given a system of equations
$$$x + 2y + 2z = {b_1}$$$
$$$5x + y + 3z = {b_2}$$$
Which of the following is true its solutions
Which of the following is true its solutions
GATE EE 2014 Set 1
17
Given that $$A = \left[ {\matrix{
{ - 5} & { - 3} \cr
2 & 0 \cr
} } \right]$$ and $${\rm I} = \left[ {\matrix{
1 & 0 \cr
0 & 1 \cr
} } \right],$$ the value of $${A^3}$$ is
GATE EE 2012
18
An eigen vector of $$p = \left[ {\matrix{
1 & 1 & 0 \cr
0 & 2 & 2 \cr
0 & 0 & 3 \cr
} } \right]$$ is
GATE EE 2010
19
The trace and determinant of a $$2 \times 2$$ matrix are shown to be $$-2$$ and $$-35$$ respectively. Its eigen values are
GATE EE 2009
20
The characteristic equation of a $$3\,\, \times \,\,3$$ matrix $$P$$ is defined as
$$\alpha \left( \lambda \right) = \left| {\lambda {\rm I} - P} \right| = {\lambda ^3} + 2\lambda + {\lambda ^2} + 1 = 0.$$
If $${\rm I}$$ denotes identity matrix then the inverse of $$P$$ will be
$$\alpha \left( \lambda \right) = \left| {\lambda {\rm I} - P} \right| = {\lambda ^3} + 2\lambda + {\lambda ^2} + 1 = 0.$$
If $${\rm I}$$ denotes identity matrix then the inverse of $$P$$ will be
GATE EE 2008
21
$$A$$ is $$m$$ $$x$$ $$n$$ full rank matrix with $$m > n$$ and $${\rm I}$$ is an identity matrix. Let matrix $${A^ + } = {\left( {{A^T}A} \right)^{ - 1}}{A^T}.$$ Then which one of the following statement is false?
GATE EE 2008
22
$$X = {\left[ {\matrix{
{{x_1}} & {{x_2}} & {.......\,{x_n}} \cr
} } \right]^T}$$ is an $$n$$-tuple non-
zero vector. The $$n\,\, \times \,\,n$$ matrix $$V = X{X^T}$$
zero vector. The $$n\,\, \times \,\,n$$ matrix $$V = X{X^T}$$
GATE EE 2007
23
In the matrix equation $$PX=Q$$ which of the following is a necessary condition for the existence of atleast one solution for the unknown vector $$X.$$
GATE EE 2005
24
The determinant of the matrix $$\left[ {\matrix{
1 & 0 & 0 & 0 \cr
{100} & 1 & 0 & 0 \cr
{100} & {200} & 1 & 0 \cr
{100} & {200} & {300} & 1 \cr
} } \right]$$ is
GATE EE 2002
25
If $$A = \left[ {\matrix{
1 & { - 2} & { - 1} \cr
2 & 3 & 1 \cr
0 & 5 & { - 2} \cr
} } \right]$$ and $$adj (A)$$ $$ = \left[ {\matrix{
{ - 11} & { - 9} & 1 \cr
4 & { - 2} & { - 3} \cr
{10} & k & 7 \cr
} } \right]$$ then $$k=$$
GATE EE 1999
26
Find the eigen values and eigen vectors of the matrix $$\left[ {\matrix{
3 & { - 1} \cr
{ - 1} & 3 \cr
} } \right]$$
GATE EE 1999
27
$$A = \left[ {\matrix{
2 & 0 & 0 & { - 1} \cr
0 & 1 & 0 & 0 \cr
0 & 0 & 3 & 0 \cr
{ - 1} & 0 & 0 & 4 \cr
} } \right].$$ The sum of the eigen values of the matrix $$A$$ is
GATE EE 1998
28
If the vector $$\left[ {\matrix{
1 \cr
2 \cr
{ - 1} \cr
} } \right]$$ is an eigen vector of $$A = \left[ {\matrix{
{ - 2} & 2 & { - 3} \cr
2 & 1 & { - 6} \cr
{ - 1} & { - 2} & 0 \cr
} } \right]$$ then one of the eigen value of $$A$$ is
GATE EE 1998
29
If $$A = \left[ {\matrix{
5 & 0 & 2 \cr
0 & 3 & 0 \cr
2 & 0 & 1 \cr
} } \right]$$ then $${A^{ - 1}} = $$
GATE EE 1998
30
A set of linear equations is represented by the matrix equations $$Ax=b.$$ The necessary condition for the existence of a solution for this system is
GATE EE 1998
31
Express the given matrix $$A = \left[ {\matrix{
2 & 1 & 5 \cr
4 & 8 & {13} \cr
6 & {27} & {31} \cr
} } \right]$$
as a product of triangular matrices $$L$$ and $$U$$ where the diagonal elements of the lower triangular matrices $$L$$ are unity and $$U$$ is an upper triangular matrix.
as a product of triangular matrices $$L$$ and $$U$$ where the diagonal elements of the lower triangular matrices $$L$$ are unity and $$U$$ is an upper triangular matrix.
GATE EE 1997
32
The inverse of the matrix $$S = \left[ {\matrix{
1 & { - 1} & 0 \cr
1 & 1 & 1 \cr
0 & 0 & 1 \cr
} } \right]$$ is
GATE EE 1995
33
Given the matrix $$A = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
{ - 6} & { - 11} & { - 6} \cr
} } \right].\,\,$$ Its eigen values are
GATE EE 1995
34
The rank of the following $$(n+1)$$ $$x$$ $$(n+1)$$ matrix, where $$'a'$$ is a real number is
$$$\left[ {\matrix{
1 & a & {{a^2}} & . & . & . & {{a^n}} \cr
1 & a & {{a^2}} & . & . & . & {{a^n}} \cr
. & {} & {} & {} & {} & {} & {} \cr
. & {} & {} & {} & {} & {} & {} \cr
1 & a & {{a^2}} & . & . & . & {{a^n}} \cr
} } \right]$$$
GATE EE 1995
35
The eigen values of the matrix $$\left[ {\matrix{
a & 1 \cr
a & 1 \cr
} } \right]$$ are
GATE EE 1994
36
$$A$$ $$\,\,5 \times 7$$ matrix has all its entries equal to $$1.$$ Then the rank of a matrix is
GATE EE 1994
37
The number of linearly independent solutions of the system of equations
$$\left[ {\matrix{ 1 & 0 & 2 \cr 1 & { - 1} & 0 \cr 2 & { - 2} & 0 \cr } } \right]\,\,\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right] = 0$$ is equal to
$$\left[ {\matrix{ 1 & 0 & 2 \cr 1 & { - 1} & 0 \cr 2 & { - 2} & 0 \cr } } \right]\,\,\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right] = 0$$ is equal to
GATE EE 1994
Marks 2
1
e4 denotes the exponential of a square matrix A. Suppose $$\lambda$$ is an eigen value and v is the corresponding eigen-vector of matrix A.
Consider the following two statements:
Statement 1 : e$$\lambda$$ is an eigen value of eA.
Statement 2 : v is an eigen-vector of eA.
Which one of the following options is correct?
GATE EE 2022
2
Consider a matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 2} \cr 0 & 1 & 1 \cr } } \right]$$. The matrix A satisfies the equation 6A$$-$$1 = A2 + cA + dI, where c and d are scalars and I is the identify matrix. Then (c + d) is equal to
GATE EE 2022
3
The eigen values of the matrix given below are $$\left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
0 & { - 3} & { - 4} \cr
} } \right]$$
GATE EE 2017 Set 2
4
Let $$P = \left[ {\matrix{
3 & 1 \cr
1 & 3 \cr
} } \right].$$ Consider the set $$S$$ of all vectors $$\left( {\matrix{
x \cr
y \cr
} } \right)$$ such that $${a^2} + {b^2} = 1$$ where $$\left( {\matrix{
a \cr
b \cr
} } \right) = P\left( {\matrix{
x \cr
y \cr
} } \right).$$ Then $$S$$ is
GATE EE 2016 Set 2
5
Let $$A$$ be a $$4 \times 3$$ real matrix which rank$$2.$$ Which one of the following statement is TRUE?
GATE EE 2016 Set 1
6
Let the eigenvalues of a $$2 \times 2$$ matrix $$A$$ be $$1,-2$$ with eigenvectors $${x_1}$$ and $${x_2}$$ respectively. Then the eigenvalues and eigenvectors of the matrix $${A^2} - 3A + 4{\rm I}$$ would respectively, be
GATE EE 2016 Set 1
7
The maximum value of $$'a'$$ such that the matrix $$\left[ {\matrix{
{ - 3} & 0 & { - 2} \cr
1 & { - 1} & 0 \cr
0 & a & { - 2} \cr
} } \right]$$ has three linearly independent real eigenvectors is
GATE EE 2015 Set 1
8
$$A = \left[ {\matrix{
p & q \cr
r & s \cr
} } \right];B = \left[ {\matrix{
{{p^2} + {q^2}} & {pr + qs} \cr
{pr + qs} & {{r^2} + {s^2}} \cr
} } \right]$$
If the rank of matrix $$A$$ is $$N$$, then the rank of matrix $$B$$ is
If the rank of matrix $$A$$ is $$N$$, then the rank of matrix $$B$$ is
GATE EE 2014 Set 3
9
A system matrix is given as follows
$$$A = \left[ {\matrix{
0 & 1 & { - 1} \cr
{ - 6} & { - 11} & 6 \cr
{ - 6} & { - 11} & 5 \cr
} } \right].$$$
The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is ___________.
GATE EE 2014 Set 1
10
The equation $$\left[ {\matrix{
2 & { - 2} \cr
1 & { - 1} \cr
} } \right]\left[ {\matrix{
{{x_1}} \cr
{{x_2}} \cr
} } \right] = \left[ {\matrix{
0 \cr
0 \cr
} } \right]$$ has
GATE EE 2013
11
A matrix has eigen values $$-1$$ and $$-2.$$ The corresponding eigenvectors are $$\left[ {\matrix{
1 \cr
{ - 1} \cr
} } \right]$$ and $$\left[ {\matrix{
1 \cr
{ - 2} \cr
} } \right]$$ respectively. The matrix is
GATE EE 2013
12
The two vectors $$\left[ {\matrix{
1 & 1 & 1 \cr
} } \right]$$ and $$\left[ {\matrix{
1 & a & {{a^2}} \cr
} } \right]$$ where $$a = - {1 \over 2} + j{{\sqrt 3 } \over 2}$$ and $$j = \sqrt { - 1} $$ are
GATE EE 2011
13
The matrix $$\left[ A \right] = \left[ {\matrix{
2 & 1 \cr
4 & { - 1} \cr
} } \right]$$ is decomposed into a product of lower triangular matrix $$\left[ L \right]$$ and an upper triangular $$\left[ U \right].$$ The properly decomposed $$\left[ L \right]$$ and $$\left[ U \right]$$ matrices respectively are
GATE EE 2011
14
For the set of equations
$$${x_1} + 2{x_2} + {x_3} + 4{x_4} = 2,$$$
$$$3{x_1} + 6{x_2} + 3{x_3} + 12{x_4} = 6.$$$
The following statement is true
The following statement is true
GATE EE 2010
15
If the rank of a $$5x6$$ matrix $$Q$$ is $$4$$ then which one of the following statements is correct?
GATE EE 2008
16
Let $$P$$ be $$2x2$$ real orthogonal matrix and $$\overline x $$ is a real vector $${\left[ {\matrix{
{{x_1}} & {{x_2}} \cr
} } \right]^T}$$ with length $$\left| {\left| {\overline x } \right|} \right| = {\left( {{x_1}^2 + {x_2}^2} \right)^{1/2}}.$$ Then which one of the following statement is correct?
GATE EE 2008
17
$${q_1},\,{q_2},{q_3},.......{q_m}$$ are $$n$$-dimensional vectors with $$m < n.$$ This set of vectors is linearly dependent. $$Q$$ is the matrix with $${q_1},\,{q_2},{q_3},.......{q_m}$$ as the columns. The rank of $$Q$$ is
GATE EE 2007
18
If $$A = \left[ {\matrix{
{ - 3} & 2 \cr
{ - 1} & 0 \cr
} } \right]\,$$ then $${A^9}$$ equals
GATE EE 2007
19
If $$A = \left[ {\matrix{
{ - 3} & 2 \cr
{ - 1} & 0 \cr
} } \right]$$ then $$A$$ satisfies the relation
GATE EE 2007
20
Let $$x$$ and $$y$$ be two vectors in a $$3-$$ dimensional space and $$ < x,y > $$ denote their dot product. Then the determinant det $$\left[ {\matrix{
{ < x,x > } & { < x,y > } \cr
{ < y,x > } & { < y,y > } \cr
} } \right] = $$ ______.
GATE EE 2007
21
If $$R = \left[ {\matrix{
1 & 0 & { - 1} \cr
2 & 1 & { - 1} \cr
2 & 3 & 2 \cr
} } \right]$$ then the top row of $${R^{ - 1}}$$ is
GATE EE 2005
22
For the matrix $$P = \left[ {\matrix{
3 & { - 2} & 2 \cr
0 & { - 2} & 1 \cr
0 & 0 & 1 \cr
} } \right],$$ one of the eigen values is $$-2.$$ Which of the following is an eigen vector?
GATE EE 2005