Linear Algebra - Mathematics Foundation - Environmental Science & Engineering Previous Year Questions

1

2024 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2024

The eigen values of the matrix \(\begin{bmatrix} 4 & 3 \\ 3 & 4 \end{bmatrix}\) are

A

1

B

2

C

7

D

4

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2

2024 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2024

If \(\mathbf{X}\) is a vector, and \(\mathbf{A}\) and \(\mathbf{B}\) are linear operators; then the correct mathematical relationship(s) is/are

A

\((\mathbf{A}+\mathbf{B})\mathbf{X} = \mathbf{A}\mathbf{X} + \mathbf{B}\mathbf{X}\)

B

\((\lambda\mathbf{A})\mathbf{X} = \lambda(\mathbf{A}\mathbf{X})\)

C

\((\mathbf{A}\mathbf{B})\mathbf{X} = \mathbf{A}(\mathbf{B}\mathbf{X})\)

D

\((\mathbf{A}+\mathbf{B})\mathbf{X} = \mathbf{A}^T\mathbf{X}+\mathbf{B}^T\mathbf{X}\)

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3

2025 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2025

The following is a system of linear equations \[ x - 2y + z = 34 \] \[ 2x + y + z = 102 \] \[ x + y - 3z = 17 \] The value of \( x + y + z \) is _________. (rounded off to two decimal places)

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4

2025 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2025

A street sweeping machine starts from point P and ends at S, as shown in the network of streets below. It sweeps all the streets at least once.

Length of the streets, in km, are shown on the network. The minimum distance travelled by the sweeping machine for completing the job of sweeping all the streets is ______ km. (rounded off to nearest integer)

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5

2021 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2021

The \(2\times2\) matrices P and Q satisfy the following relations: \(P + Q = \begin{bmatrix} 3 & 1 \\ 2 & 12 \end{bmatrix}\) and \(P - Q = \begin{bmatrix} -1 & -7 \\ 8 & 2 \end{bmatrix}\) The matrix Q is equal to __________.

A

\(\begin{bmatrix} 2 & 4 \\ -3 & 5 \end{bmatrix}\)

B

\(\begin{bmatrix} 1 & -3 \\ 5 & 7 \end{bmatrix}\)

C

\(\begin{bmatrix} 2 & -3 \\ 4 & 5 \end{bmatrix}\)

D

\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

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6

2022 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2022

Given P is m × n matrix, Q is n × l matrix, R and S are n × n matrices, and superscript T denotes transpose. Consider
Relationship 1: (PQ)^T = Q^TP^T
Relationship 2: (RS)^-1 = S^-1R^-1
Which one of the following is correct?

A

Relationship 1 is false; Relationship 2 is false

B

Relationship 1 is true; Relationship 2 is false

C

Relationship 1 is true; Relationship 2 is true

D

Relationship 1 is false; Relationship 2 is true

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7

2022 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2022

Eigen values of the matrix \( \begin{bmatrix} 0 & 1 \\ -2 & 3 \end{bmatrix} \) are

A

1 and 2

B

1 and 3

C

1 and –2

D

0 and 3

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8

2023 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2023

P and Q are square matrices. Consider the following
X: \((P^{-1})^{-1} = P\)
Y: Symmetric if \(Q = -Q^T\)
The correct choice is

A

X is TRUE; Y is FALSE

B

X is FALSE; Y is TRUE

C

Both X and Y are TRUE

D

Both X and Y are FALSE

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9

2023 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2023

Consider two matrices \(\mathbf{P} = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}\) and \(\mathbf{Q} = \begin{bmatrix} 5 & 4 \\ 0 & 2 \end{bmatrix}\). If \(\mathbf{R} = (\mathbf{P}\mathbf{Q})^T\) then det \(\mathbf{R}\) is ______________ (in integer).

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10

2026 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2026

If \(2X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}\), then \(X\) is ______.

A

\(\begin{bmatrix} 2 & 5 \\ 5 & 3 \end{bmatrix}\)

B

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

C

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

D

\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

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11

2026 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2026

Consider the matrix \[ A = \begin{bmatrix} 2 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 4 & 10 \end{bmatrix} \].
Which ONE of the following options CORRECTLY lists the eigenvalues of \( A \)?

A

1; 8 – √37; –8 + √37

B

2; 8 – √37; –8 + √37

C

1; 8 + √37; 8 – √37

D

2; 8 + √37; 8 + √37

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12

2026 · Environmental Science & Engineering · Mathematics Foundation · Linear Algebra

Environmental Science & Engineering (ES) 2026

A system of linear equations is given as \(AX = B\), where \(A = \begin{bmatrix} 2 & 5 \\ 5 & -2 \end{bmatrix}\), \(B = \begin{bmatrix} 7 \\ 3 \end{bmatrix}\), and \(X\) is the solution vector. Which of the following statements is/are TRUE for this system?

A

Matrix \(A\) is skew-symmetric.

B

Eigenvalues of matrix \(A\) are real.

C

The system is homogeneous.

D

The system is singular.

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