Linear Algebra - Engineering Mathematics - Petroleum Engineering Previous Year Questions

1

2016 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2016

Consider the matrix, \(\mathbf{M} = \begin{bmatrix} 5 & 3 \\ 3 & 5 \end{bmatrix}\). The normalized eigen-vector corresponding to the smallest eigen-value of the matrix \(\mathbf{M}\) is

A

\(\begin{pmatrix} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{pmatrix}\)

B

\(\begin{pmatrix} \frac{\sqrt{3}}{2} \\ -\frac{1}{2} \end{pmatrix}\)

C

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

D

\(\begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}\)

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2

2017 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2017

For the two matrices \( X = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \), \( Y = \begin{bmatrix} 7 & 0 \\ 8 & -1 \end{bmatrix} \), the product \( XY \) will be:

A

\(\begin{bmatrix} 7 & 14 & 21 \\ 4 & 11 & 18 \end{bmatrix} \)

B

\(\begin{bmatrix} 4 & 11 & 18 \\ 7 & 14 & 21 \end{bmatrix} \)

C

\(\begin{bmatrix} 7 & 14 & 18 \\ 14 & 11 & 21 \end{bmatrix} \)

D

\(\begin{bmatrix} 7 & 14 & 21 \\ 18 & 5 & 6 \end{bmatrix} \)

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3

2017 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2017

The eigenvalues for the matrix \(\begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}\) are:

A

2 and 5

B

-2 and -5

C

-2 and 5

D

none of the above

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4

2018 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2018

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

A

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

B

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

C

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

D

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

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5

2019 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2019

For any real, square and non-singular matrix B, the det B^-1 is

A

zero

B

(det B)^-1

C

-(det B)

D

det B

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6

2019 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2019

An LPP is defined as
Minimize \( z = 15x_1 + 12x_2 \)
subject to,
\( x_1 + 2x_2 \leq 3 \)
\( 2x_1 - 4x_2 \leq 5 \)
\( x_1, x_2 \geq 0 \)
The objective function of the dual of this LPP is

A

Maximize \(w = y_1 + y_2 \)

B

Maximize \(w = y_1 + 2y_2 \)

C

Maximize \(w = 2y_1 - 4y_2 \)

D

Maximize \(w = 3y_1 + 5y_2 \)

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7

2020 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2020

An incompressible fluid flows through a network of pipes as shown in the given Figure. The total pressure drop across points \(a\) and \(b\) is 2 kPa. The flow rates (in m³/s) in sections 1, 2, and 3 are \(q_1\), \(q_2\), and \(q_3\) respectively. The pressure drops (in kPa) are \(4q_1\), \(3q_2\), and \(2q_3\) across sections 1, 2, and 3, respectively.

For a steady-state flow operation, the system of equations for flow rates is given by,
\[ \begin{bmatrix} 4 & 3 & 0 \\ 0 & -3 & 2 \\ 0 & X & -1 \end{bmatrix} \begin{bmatrix} q_1 \\ q_2 \\ q_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ -0.5 \end{bmatrix} \]

The correct option for the numeric value of \(X\) is

A

-0.50

B

-1.75

C

-1.00

D

-2.00

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8

2020 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2020

An incompressible fluid flows through a network of pipes as shown in the given Figure. The total pressure drop across points $a$ and $b$ is 2 kPa. The flow rates (in m³/s) in sections 1, 2, and 3 are $q_1, q_2$, and $q_3$ respectively. The pressure drops (in kPa) are $4q_1, 3q_2$, and $2q_3$ across sections 1, 2, and 3, respectively. For a steady-state flow operation, the system of equations for flow rates is given by, \[ \begin{bmatrix} 4 & 3 & 0 \\ 0 & -3 & 2 \\ 0 & X & -1 \end{bmatrix} \begin{bmatrix} q_1 \\ q_2 \\ q_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ -0.5 \end{bmatrix} \] The correct option for the numeric value of $X$ is

A

-0.50

B

-1.75

C

-1.00

D

-2.00

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9

2021 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2021

Identify the CORRECT statements for a n × n matrix.

A

Under elementary row operations, the rank of the matrix remains invariant

B

Under elementary row operations, the eigenvalues of the matrix remain the same

C

If the elements in a row can be written as a linear combination of two or more rows, then the matrix is singular.

D

The rank of the matrix is equal to n if the determinant of the matrix is zero.

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10

2021 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2021

Given matrix $A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}$. The eigenvalue corresponding to the eigenvector $\begin{bmatrix} 1 \\ -1 \end{bmatrix}$ is __________.

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11

2022 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2022

Which of the following properties of Matrix \(\mathbf{A} = \begin{bmatrix} 1 & 0.5 & 0 \\ 0.5 & 1 & 0.5 \\ 0 & 0.5 & 1 \end{bmatrix}\) are CORRECT?

A

Singular

B

Positive definite

C

Symmetric

D

Diagonal

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12

2024 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2024

For the given matrix \( Q = \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 1 & 0 \\ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{bmatrix} \), which of the following statements is/are true?

A

\(Q \) is an orthogonal matrix

B

\(Q^T = Q^{-1} \)

C

\(Q \) is a singular matrix

D

\(Q \) is a symmetric matrix

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13

2024 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2024

If \( P = \begin{bmatrix} 2 & -1 \\ 2 & 2 \end{bmatrix} \), the product of the eigenvalues of \( P \) is ________.

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14

2025 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2025

The eigenvalues of the matrix \(\begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix}\) are \(\lambda_1, \lambda_2\), and \(\lambda_3\).
The value of \(\lambda_1 \lambda_2 \lambda_3 (\lambda_1 + \lambda_2 + \lambda_3)\) is

A

11

B

45

C

396

D

495

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15

2023 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2023

Let \mathbf{X} = \begin{bmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \end{bmatrix} be a 3 × 3 matrix.
The determinant of matrix \mathbf{X} is 5.
The determinant of matrix \mathbf{Y} = \begin{bmatrix} x_{11} & x_{12} & x_{13} \\ 2x_{21} & 2x_{22} & 2x_{23} \\ 3x_{31} & 3x_{32} & 3x_{33} \end{bmatrix} is ______.

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16

2026 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2026

Which of the following options is CORRECT for the eigenvalues (λ) of the given matrix? \[ \begin{bmatrix} 8 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix} \]

A

λ = 1, 1, 1

B

λ = 8, 5, 7

C

λ = 0, 0, 0

D

λ = 40, 1, 35

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17

2026 · Petroleum Engineering · Engineering Mathematics · Linear Algebra

Petroleum Engineering (PE) 2026

For a given matrix \( A = \begin{bmatrix} 4 & 4 \\ 0 & 4 \\ -4 & 4 \end{bmatrix} \), and \( A^T \) representing its transpose, which of the following options is the CORRECT representation of \( A^T A \)?

A

\(\begin{bmatrix} 4 & 0 \\ 0 & -4 \end{bmatrix} \)

B

\(\begin{bmatrix} 32 & 0 \\ 0 & 48 \end{bmatrix} \)

C

\(\begin{bmatrix} 32 & 0 \\ 0 & 0 \\ 0 & 48 \end{bmatrix} \)

D

\(\begin{bmatrix} 4 & 0 \\ 0 & 0 \\ 0 & -4 \end{bmatrix} \)

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