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Linear Algebra · Engineering Mathematics · GATE CE

Marks 1
47
Marks 2
16
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Marks 1

1

The statements P and Q are related to matrices A and B, which are conformable for both addition and multiplication.

P: $(A + B)^T = A^T + B^T$

Q: $(AB)^T = B^T A^T$

Which one of the following options is CORRECT?

GATE CE 2024 Set 2
2

For the matrix

$\rm [A]=\begin{bmatrix}1&-1&0\\\ -1&2&-1\\\ 0&-1&1\end{bmatrix}$

which of the following statements is/are TRUE?

GATE CE 2023 Set 2
3
If M is an arbitrary real n × n matrix, then which of the following matrices will have non-negative eigenvalues? 
GATE CE 2023 Set 1
4

Let y be a non-zero vector of size 2022 $$\times$$ 1. Which of the following statements is/are TRUE?

GATE CE 2022 Set 2
5

The components of pure shear strain in a sheared are given in the matrix form:

$$\varepsilon = \left[ {\matrix{ 1 & 1 \cr 1 & { - 1} \cr } } \right]$$

Here, Trace ($$\varepsilon $$) = 0. Given, P = Trace ($$\varepsilon$$8) and Q = Trace ($$\varepsilon $$11).

The numerical value of (P + Q) is ___________. (in integer)

GATE CE 2022 Set 2
6

P and Q are two square matrices of the same order. Which of the following statements is/are correct?

GATE CE 2022 Set 2
7

The matrix M is defined as

$$M = \left[ {\matrix{ 1 & 3 \cr 4 & 2 \cr } } \right]$$

and has eigenvalues 5 and $$-$$2. The matrix Q is formed as

Q = M3 $$-$$ 4M2 $$-$$ 2M

Which of the following is/are the eigenvalue(s) of matrix Q?

GATE CE 2022 Set 1
8

The Cartesian coordinates of a point P in a right-handed coordinate system are (1, 1, 1). The transformed coordinates of P due to a 45$$^\circ$$ clockwise rotation of the coordinate system about the positive x-axis are

GATE CE 2022 Set 1
9
Consider the following simultaneous equations (with $${c_1}$$ and $${c_2}$$ being constants): $$$3{x_1} + 2{x_2} = {c_1}$$$ $$$4{x_1} + {x_2} = {c_2}$$$

The characteristic equation for these simultaneous equation is

GATE CE 2017 Set 2
10
The matrix $$P$$ is the inverse of a matrix $$Q.$$ If $${\rm I}$$ denotes the identity matrix, which one of the following options is correct?
GATE CE 2017 Set 1
11
If the entries in each column of a square matrix $$M$$ add up to $$1$$, then an eigenvalue of $$M$$ is
GATE CE 2016 Set 1
12
For what value of $$'p'$$ the following set of equations will have no solutions? $$$2x+3y=5$$$ $$$3x+py=10$$$
GATE CE 2015 Set 1
13
Let $$A = \left[ {{a_{ij}}} \right],\,\,1 \le i,j \le n$$ with $$n \ge 3$$ and $${{a_{ij}} = i.j.}$$
The rank of $$A$$ is :
GATE CE 2015 Set 2
14
The sum of Eigen values of the matrix, $$\left[ M \right]$$
is where $$\left[ M \right] = \left[ {\matrix{ {215} & {650} & {795} \cr {655} & {150} & {835} \cr {485} & {355} & {550} \cr } } \right]$$
GATE CE 2014 Set 1
15
Given the matrices $$J = \left[ {\matrix{ 3 & 2 & 1 \cr 2 & 4 & 2 \cr 1 & 2 & 6 \cr } } \right]$$ and $$K = \left[ {\matrix{ 1 \cr 2 \cr { - 1} \cr } } \right],\,\,$$ the product $${K^T}JK$$ is ______.
GATE CE 2014 Set 1
16
The determinant of matrix $$\left[ {\matrix{ 0 & 1 & 2 & 3 \cr 1 & 0 & 3 & 0 \cr 2 & 3 & 0 & 1 \cr 3 & 0 & 1 & 2 \cr } } \right]$$ is
GATE CE 2014 Set 2
17
The rank of the matrix $$\left[ {\matrix{ 6 & 0 & 4 & 4 \cr { - 2} & {14} & 8 & {18} \cr {14} & { - 14} & 0 & { - 10} \cr } } \right]$$ is
GATE CE 2014 Set 2
18
The eigen values of matrix $$\left[ {\matrix{ 9 & 5 \cr 5 & 8 \cr } } \right]$$ are
GATE CE 2012
19
A square matrix $$B$$ is symmetric if ____
GATE CE 2009
20
In the solution of the following set of linear equations by Gauss-elimination using partial pivoting $$$5x+y+2z=34,$$$ $$$4y-3z=12$$$ and $$$10x-2y+z=-4.$$$
The pivots for elimination of $$x$$ and $$y$$ are
GATE CE 2009
21
The following system of equations $$$x+y+z=3,$$$ $$$x+2y+3z=4,$$$ $$$x+4y+kz=6$$$
will not have a unique solution for $$k$$ equal to
GATE CE 2008
22
The product of matrices $${\left( {PQ} \right)^{ - 1}}P$$ is
GATE CE 2008
23
The eigenvalues of the matrix $$\left[ P \right] = \left[ {\matrix{ 4 & 5 \cr 2 & { - 5} \cr } } \right]$$ are
GATE CE 2008
24
Solution for the system defined by the set of equations $$4y+3z=8, 2x-z=2$$ & $$3x+2y=5$$ is
GATE CE 2006
25
Consider the following system of equations in three real variable $${x_1},$$ $${x_2}$$ and $${x_3}:$$ $$$2{x_1} - {x_2} + 3{x_3} = 1$$$ $$$3{x_1} + 2{x_2} + 5{x_3} = 2$$$ $$$ - {x_1} + 4{x_2} + {x_3} = 3$$$

This system of equations has

GATE CE 2005
26
Consider a non-homogeneous system of linear equations represents mathematically an over determined system. Such a system will be
GATE CE 2005
27
Consider the matrices $$\,{X_{4x3,}}\,\,{Y_{4x3}}$$ $$\,\,{P_{2x3}}.$$ The order of $$\,{\left[ {P{{\left( {{X^T}Y} \right)}^{ - 1}}{P^T}} \right]^T}$$ will be
GATE CE 2005
28
Consider the system of equations, $${A_{nxn}}\,\,{X_{nx1}}\,\, = \lambda \,{X_{nx1}}$$ where $$\lambda $$ is a scalar. Let $$\left( {{\lambda _i},\,\,{X_i}} \right)$$ be an eigen value and its corresponding eigen vector for real matrix $$A$$. Let $${{\rm I}_{nxn}}$$ be unit matrix. Which one of the following statement is not correct?
GATE CE 2005
29
Real matrices $$\,\,{\left[ A \right]_{3x1,}}$$ $$\,\,{\left[ B \right]_{3x3,}}$$ $$\,\,{\left[ C \right]_{3x5,}}$$ $$\,\,{\left[ D \right]_{5x3,}}$$ $$\,\,{\left[ E \right]_{5x5,}}$$ $$\,\,{\left[ F \right]_{5x1,}}$$ are given. Matrices $$\left[ B \right]$$ and $$\left[ E \right]$$ are symmetric. Following statements are made with respect to their matrices.
$$(I)$$ Matrix product $$\,\,{\left[ F \right]^T}\,\,$$ $$\,\,{\left[ C \right]^T}\,\,$$ $$\,\,\left[ B \right]\,\,$$ $$\,\,\left[ C \right]\,\,$$ $$\,\,\left[ F \right]\,\,$$ is a scalar.
$$(II)$$ Matrix product $$\,\,{\left[ D \right]^T}\,\,$$ $$\,\left[ F \right]\,\,$$ $$\,\left[ D \right]\,\,$$ is always symmetric.
With reference to above statements which of the following applies?
GATE CE 2004
30
The eigen values of the matrix $$\left[ {\matrix{ 4 & { - 2} \cr { - 2} & 1 \cr } } \right]$$ are
GATE CE 2004
31
Given matrix $$\left[ A \right] = \left[ {\matrix{ 4 & 2 & 1 & 3 \cr 6 & 3 & 4 & 7 \cr 2 & 1 & 0 & 1 \cr } } \right],$$ the rank of the matrix is
GATE CE 2003
32
Eigen values of the following matrix are $$\left[ {\matrix{ { - 1} & 4 \cr 4 & { - 1} \cr } } \right]$$
GATE CE 2002
33
The determinant of the following matrix $$\left[ {\matrix{ 5 & 3 & 2 \cr 1 & 2 & 6 \cr 3 & 5 & {10} \cr } } \right]$$
GATE CE 2001
34
The product $$\left[ P \right]\,\,{\left[ Q \right]^T}$$ of the following two matrices $$\left[ P \right]\,$$ and $$\left[ Q \right]\,$$
where $$\left[ P \right]\,\, = \left[ {\matrix{ 2 & 3 \cr 4 & 5 \cr } } \right],\,\,\left[ Q \right] = \left[ {\matrix{ 4 & 8 \cr 9 & 2 \cr } } \right]$$ is
GATE CE 2001
35
The eigen values of the matrix $$\left[ {\matrix{ 5 & 3 \cr 2 & 9 \cr } } \right]$$ are
GATE CE 2001
36
Consider the following two statements.

$$(I)$$ The maximum number of linearly independent column vectors of a matrix $$A$$ is called the rank of $$A.$$

$$(II)$$ If $$A$$ is $$nxn$$ square matrix then it will be non-singular if rank of $$A=n$$

GATE CE 2000
37
If $$A,B,C$$ are square matrices of the same order then $${\left( {ABC} \right)^{ - 1}}$$ is equal be
GATE CE 2000
38
If $$A$$ is any $$nxn$$ matrix and $$k$$ is a scalar then $$\left| {kA} \right| = \alpha \left| A \right|$$ where $$\alpha $$ is
GATE CE 1999
39
The equation $$\left[ {\matrix{ 2 & 1 & 1 \cr 1 & 1 & { - 1} \cr y & {{x^2}} & x \cr } } \right] = 0$$ represents a parabola passing through the points.
GATE CE 1999
40
The number of terms in the expansion of general determinant of order $$n$$ is
GATE CE 1999
41
In matrix algebra $$AS=AT$$ ($$A,S,T,$$ are matrices of appropriate order) implies $$S=T$$ only if
GATE CE 1998
42
The real symmetric matrix $$C$$ corresponding to the quadratic form $$Q = 4{x_1}{x_2} - 5{x_2}{x_2}$$ is
GATE CE 1998
43
Obtain the eigen values and eigen vectors of $$A = \left[ {\matrix{ 8 & -4 \cr 2 & { 2 } \cr } } \right].$$
GATE CE 1998
44
If $$A$$ is a real square matrix then $$A{A^T}$$ is
GATE CE 1998
45
Inverse of matrix $$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \cr } } \right]$$ is
GATE CE 1997
46
If $$A$$ and $$B$$ are two matrices and $$AB$$ exists then $$BA$$ exists,
GATE CE 1997
47
If the determinant of the matrix $$\left[ {\matrix{ 1 & 3 & 2 \cr 0 & 5 & { - 6} \cr 2 & 7 & 8 \cr } } \right]$$ is $$26,$$ then the determinant of
the matrix $$\left[ {\matrix{ 2 & 7 & 8 \cr 0 & 5 & { - 6} \cr 1 & 3 & 2 \cr } } \right]$$ is
GATE CE 1997

Marks 2

1

Consider two matrices $A = \begin{bmatrix}2 & 1 & 4 \\ 1 & 0 & 3\end{bmatrix}$ and $B = \begin{bmatrix}-1 & 0 \\ 2 & 3 \\ 1 & 4 \end{bmatrix}$.

The determinant of the matrix $AB$ is __________ (in integer).

GATE CE 2024 Set 2
2

What are the eigenvalues of the matrix $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ ?

GATE CE 2024 Set 1
3
Two vectors [2 1 0 3]𝑇 and [1 0 1 2]𝑇 belong to the null space of a 4 × 4 matrix of rank 2. Which one of the following vectors also belongs to the null space?
GATE CE 2023 Set 2
4

Cholesky decomposition is carried out on the following square matrix [𝐴]. 

$\rm [A]=\begin{bmatrix}8&-5\\\ -5&a_{22}\end{bmatrix}$

Let 𝑙ij and 𝑎ij be the (i, j)th elements of matrices [𝐿] and [𝐴], respectively. If the element 𝑙22 of the decomposed lower triangular matrix [𝐿] is 1.968, what is the value (rounded off to the nearest integer) of the element 𝑎22?

GATE CE 2023 Set 2
5

For the matrix

$[A]= \begin{bmatrix}1&2&3\\\ 3&2&1\\\ 3&1&2 \end{bmatrix} $

which of the following statements is/are TRUE?

GATE CE 2023 Set 1
6
If $$A = \left[ {\matrix{ 1 & 5 \cr 6 & 2 \cr } } \right]\,\,and\,\,B = \left[ {\matrix{ 3 & 7 \cr 8 & 4 \cr } } \right]A{B^T}$$ is equal to
GATE CE 2017 Set 2
7
Consider the matrix $$\left[ {\matrix{ 5 & { - 1} \cr 4 & 1 \cr } } \right].$$ Which one of the following statements is TRUE for the eigenvalues and eigenvectors of this matrix?
GATE CE 2017 Set 1
8
Consider the following linear system $$$x+2y-3z=a$$$ $$$2x+3y+3z=b$$$ $$$5x+9y-6z=c$$$
This system is consistent if $$a,b$$ and $$c$$ satisfy the equation
GATE CE 2016 Set 2
9
The smallest and largest Eigen values of the
following matrix are : $$\left[ {\matrix{ 3 & { - 2} & 2 \cr 4 & { - 4} & 6 \cr 2 & { - 3} & 5 \cr } } \right]$$
GATE CE 2015 Set 1
10
The two Eigen Values of the matrix $$\left[ {\matrix{ 2 & 1 \cr 1 & p \cr } } \right]$$ have a ratio of $$3:1$$ for $$p=2.$$ What is another value of $$'p'$$ for which the Eigen values have the same ratio of $$3:1$$?
GATE CE 2015 Set 2
11
What is the minimum number of multiplications involved in computing the matrix product $$PQR?$$ Matrix $$P$$ has $$4$$ rows and $$2$$ columns, matrix $$Q$$ has $$2$$ rows and $$4$$ columns and matrix $$R$$ has $$4$$ rows and $$1$$ column ______.
GATE CE 2013
12
The inverse of the matrix $$\left[ {\matrix{ {3 + 2i} & i \cr { - i} & {3 - 2i} \cr } } \right]$$ is
GATE CE 2010
13
The inverse of $$2 \times 2$$ matrix $$\left[ {\matrix{ 1 & 2 \cr 5 & 7 \cr } } \right]$$ is
GATE CE 2007
14
The minimum and maximum eigen values of matrix $$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \cr } } \right]$$ are $$-2$$ and $$6$$ respectively. What is the other eigen value?
GATE CE 2007
15
For what values of $$\alpha $$ and $$\beta $$ the following simultaneous equations have an infinite number of solutions $$$x+y+z=5,$$$ $$$x+3y+3z=9,$$$ $$$x + 2y + \alpha z = \beta $$$
GATE CE 2007
16
For a given matrix $$A = \left[ {\matrix{ 2 & { - 2} & 3 \cr { - 2} & { - 1} & 6 \cr 1 & 2 & 0 \cr } } \right],$$ one of the eigen value is $$3.$$ The other two eigen values are
GATE CE 2006
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