Linear Algebra - Engineering Mathematics - Chemical Engineering Previous Year Questions

1

2007 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2007

The value of “\(a\)” for which the following set of equations
\(y + 2z = 0\)
\(2x + y + z = 0\)
\(ax + 2y = 0\)
have non-trivial solution, is

A

0

B

8

C

-2

D

3

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2

2007 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2007

\( \underline{A} \) and \( \underline{B} \) are two \( 3 \times 3 \) matrices such that
\( \underline{A} = \begin{bmatrix} -2 & 4 & 6 \\ 1 & 2 & 1 \\ 0 & 4 & 4 \end{bmatrix} \), \( \underline{B} = \underline{0} \)
and \( \underline{A}\underline{B} = \underline{0} \). Then the rank of matrix \( \underline{B} \) is

A

r = 2

B

r < 3

C

r ≤ 3

D

r = 3

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3

2008 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2008

If \( A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} \), then the eigenvalues of \( A^3 \) are

A

5, 4

B

3, -1

C

9, -1

D

27, -1

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4

2009 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2009

The system of linear equations \( \mathbf{A} \mathbf{x} = \mathbf{0} \), where \( \mathbf{A} \) is an \( n \times n \) matrix, has a non-trivial solution ONLY if

A

rank of \(\mathbf{A} > n \)

B

rank of \(\mathbf{A} = n \)

C

rank of \(\mathbf{A} < n \)

D

\(\mathbf{A} \) is an identity matrix

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5

2009 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2009

The eigenvalues of matrix \(A = \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix}\) are 5 and \(-1\). Then the eigenvalues of \(-2A + 3I\) (\(I\) is a \(2 \times 2\) identity matrix) are

A

\(-7\) and 5

B

7 and \(-5\)

C

\(-\frac{1}{7}\) and \(\frac{1}{5}\)

D

\(\frac{1}{7}\) and \(-\frac{1}{5}\)

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6

2010 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2010

The inverse of the matrix \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) is

A

\(\begin{bmatrix} -2 & -1 \\ -3/2 & -1/2 \end{bmatrix} \)

B

\(\begin{bmatrix} -2 & 3/2 \\ 1 & -1/2 \end{bmatrix} \)

C

\(\begin{bmatrix} -2 & 1 \\ 3/2 & -1/2 \end{bmatrix} \)

D

\(\begin{bmatrix} 2 & -3/2 \\ -1 & 1/2 \end{bmatrix} \)

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7

2011 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2011

Let \(\lambda_1 = -1\) and \(\lambda_2 = 3\) be the eigenvalues and \(\underline{V_1} = \begin{pmatrix}1\\0\end{pmatrix}\) and \(\underline{V_2} = \begin{pmatrix}1\\1\end{pmatrix}\) be the corresponding eigenvectors of a real \(2 \times 2\) matrix \(\underline{\underline{P}}\). Given that \(\underline{\underline{P}} = (\underline{V_1} \; \underline{V_2})\), which ONE of the following matrices represents \(\underline{\underline{P}}^{-1} \underline{\underline{R}} \underline{\underline{P}}\)?

A

\(\begin{pmatrix}0 & -1\\3 & 0\end{pmatrix}\)

B

\(\begin{pmatrix}0 & 3\\-1 & 0\end{pmatrix}\)

C

\(\begin{pmatrix}3 & 0\\0 & -1\end{pmatrix}\)

D

\(\begin{pmatrix}-1 & 0\\0 & 3\end{pmatrix}\)

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8

2011 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2011

A transponder receives the same number of orders each day. Currently, he has some pending orders (backlog) to be shipped. If he uses 7 trucks, then at the end of the 4th day he can clear all the orders. Alternatively, if he uses only 3 trucks, then all the orders are cleared at the end of the 10th day. What is the minimum number of trucks required so that there will be no pending order at the end of the 5th day?

A

4

B

5

C

6

D

7

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9

2012 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2012

Consider the following set of linear algebraic equations
\[ \begin{array}{rcl} x_1 + 2x_2 + 3x_3 &=& 2 \\ x_2 + x_3 &=& -1 \\ 2x_2 + 2x_3 &=& 0 \end{array} \]
The system has

A

a unique solution

B

no solution

C

an infinite number of solutions

D

only the trivial solution

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10

2012 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2012

Consider the following \( (2 \times 2) \) matrix
\[ \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} \]
Which one of the following vectors is NOT a valid eigenvector of the above matrix?

A

\(\begin{pmatrix} 1 \\ 0 \end{pmatrix} \)

B

\(\begin{pmatrix} -2 \\ 1 \end{pmatrix} \)

C

\(\begin{pmatrix} 4 \\ -3 \end{pmatrix} \)

D

\(\begin{pmatrix} 0 \\ 0 \end{pmatrix} \)

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11

2013 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2013

Which of the following statements are TRUE?
P. The eigenvalues of a symmetric matrix are real
Q. The value of the determinant of an orthogonal matrix can only be +1
R. The transpose of a square matrix A has the same eigenvalues as those of A
S. The inverse of an 'n x n' matrix exists if and only if the rank is less than 'n'

A

P and Q only

B

P and R only

C

Q and R only

D

P and S only

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12

2015 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2015

The following set of three vectors \[ \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \begin{pmatrix} x \\ 6 \\ x \end{pmatrix} \text{ and } \begin{pmatrix} 3 \\ 4 \\ 2 \end{pmatrix} \] is linearly dependent when x is equal to

A

0

B

1

C

2

D

3

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13

2015 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2015

For the matrix \(\begin{pmatrix} 4 & 3 \\ 3 & 4 \end{pmatrix}\), if \(\begin{pmatrix} 1 \\ 1 \end{pmatrix}\) is an eigenvector, the corresponding eigenvalue is ________.

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14

2018 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2018

For the matrix \( A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \) if \( det \) stands for the determinant and \( A^T \) is the transpose of A then the value of \( det(A^T A) \) is __________

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15

2019 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2019

The product of the eigenvalues of the matrix \(\begin{pmatrix} 2 & 3 \\ 0 & 7 \end{pmatrix}\) is __________ (rounded off to one decimal place).

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16

2019 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2019

A system of \( n \) homogeneous linear equations containing \( n \) unknowns will have non-trivial solutions if and only if the determinant of the coefficient matrix is

A

1

B

-1

C

0

D

∞

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17

2020 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2020

Sum of the eigenvalues of the matrix \(\begin{bmatrix} 2 & 4 & 6 \\ 3 & 5 & 9 \\ 12 & 1 & 7 \end{bmatrix}\) is __________ (round off to nearest integer).

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18

2021 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2021

A, B, C and D are vectors of length 4.
\[ A = [a_1 \quad a_2 \quad a_3 \quad a_4] \]
\[ B = [b_1 \quad b_2 \quad b_3 \quad b_4] \]
\[ C = [c_1 \quad c_2 \quad c_3 \quad c_4] \]
\[ D = [d_1 \quad d_2 \quad d_3 \quad d_4] \]
It is known that B is not a scalar multiple of A. Also, C is linearly independent of A and B. Further, D = 3A + 2B + C.
The rank of the matrix \[ \begin{bmatrix} a_1 & a_2 & a_3 & a_4 \\ b_1 & b_2 & b_3 & b_4 \\ c_1 & c_2 & c_3 & c_4 \\ d_1 & d_2 & d_3 & d_4 \end{bmatrix} \] is __________

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19

2021 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2021

Let \(A\) be a square matrix of size \(n \times n (n > 1)\). The elements of \(A = \{a_{ij}\}\) are given by
\(a_{ij} = \begin{cases} i \times j, & \text{if } i \geq j \\ 0, & \text{if } i < j \end{cases}\)
The determinant of \(A\) is

A

0

B

1

C

\(n !\)

D

\((n !)^{2}\)

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20

2022 · Chemical Engineering · Engineering Mathematics · Linear Algebra

Chemical Engineering (CH) 2022

Given matrix \(A = \begin{bmatrix} x & 1 & 3 \\ y & 2 & 6 \\ 3 & 5 & 7 \end{bmatrix}\), the ordered pair \((x, y)\) for which \(\det(A) = 0\) is

A

\((1, 1)\)

B

\((1, 2)\)

C

\((2, 2)\)

D

\((2, 1)\)

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