Linear Algebra - Engineering Mathematics - Naval Architecture & Marine Engineering Previous Year Questions

1

2022 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Naval Architecture & Marine Engineering (NM) 2022

Let \( A \) be a real non-zero square matrix of order \( n \). If the homogeneous system of linear equations \( Ax = 0 \) has only trivial solution, then

A

the matrix \(A \) is singular

B

the determinant of \(A \) is zero

C

\(\lambda = 0 \) is an eigenvalue of \(A \)

D

for any n-vector \(b \), the system of linear equations \(Ax = b \) has a unique solution

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2

2022 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Naval Architecture & Marine Engineering (NM) 2022

Let \( M = \begin{pmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{pmatrix} \). Which of the following are TRUE?

A

\(M \) is singular

B

\(M^{-1} = \frac{1}{4} M^2 - \frac{3}{2} M + \frac{9}{4} I \), where \(I \) is the identity matrix of order 3

C

\(M \) has three distinct eigenvalues

D

\(M \) has three linearly independent eigenvectors

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3

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Naval Architecture & Marine Engineering (NM) 2023

If the system of linear equations, \(x - ay - z = 0\), \(ax - y - z = 0\), \(x + y - z = 0\), has infinite number of solutions, then the possible values of \(a\) are

A

0, 1

B

-1, 2

C

-1, 1

D

0, -1

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4

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Naval Architecture & Marine Engineering (NM) 2023

Let \( L = \begin{pmatrix} 3 & -1 & -1 & -1 \\ -1 & 2 & -1 & 0 \\ -1 & -1 & 3 & 0 \\ -1 & 0 & -1 & 1 \end{pmatrix} \). Which of the following are TRUE?

A

The matrix \(L \) is row equivalent to \(\begin{pmatrix} 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ -1 & -1 & 3 & 0 \\ -1 & 0 & -1 & 1 \end{pmatrix} \)

B

The linear system \(Lx = b \) has a solution for all \(b \)

C

For \(b \neq \begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \end{pmatrix} \), the system \(Lx = b \) has a solution

D

Rank of the matrix \(L \) is 3

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5

2024 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Naval Architecture & Marine Engineering (NM) 2024

The system of linear equations
\(x + 2y + 3z = 4\)
\(2x - y - 2z = a^2\)
\(-x - 7y - 11z = a\)
has a solution if the values of \(a\) are

A

\(-1\) and \(5\)

B

\(-2\) and \(3\)

C

\(-5\) and \(1\)

D

\(-3\) and \(4\)

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6

2024 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Naval Architecture & Marine Engineering (NM) 2024

Consider the matrices \( M = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix} \) and \( N = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 2 & 0 \\ 1 & 1 & 0 \end{pmatrix} \). Which one of the following is true?

A

M is not diagonalizable but N is diagonalizable

B

Both M and N are not diagonalizable

C

Both M and N are diagonalizable

D

M is diagonalizable but N is not diagonalizable

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7

2025 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Naval Architecture & Marine Engineering (NM) 2025

Let \( M = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \) and \( K = \begin{bmatrix} 2 & -1 \\ -1 & 1 \end{bmatrix} \) satisfy the eigenvalue problem given by \( [M - \alpha K] \varphi = 0 \).
The lowest eigenvalue \( \alpha \) is __________ (rounded off to two decimal places).

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8

2026 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2026

If \( I \) is an identity matrix, the polynomial equation that satisfies the Cayley-Hamilton theorem for the matrix \( P = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix} \) is

A

\(P^2 - 5P + 6I = 0 \)

B

\(P^2 - 5P + 5I = 0 \)

C

\(P^2 - 6P + 5I = 0 \)

D

\(P^2 - 4P + 3I = 0 \)

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9

2026 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2026

Given $\mathbf{x} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$, $\mathbf{b} = \begin{bmatrix} -16 \\ -3 \\ 7 \end{bmatrix}$, $\mathbf{P}^{-1} = \begin{bmatrix} e & -6 & -7 \\ f & 9 & 11 \\ -1 & -2 & g \end{bmatrix}$ and $\mathbf{P}\mathbf{x} = \mathbf{b}$. If $\mathbf{P}\mathbf{y} = \mathbf{d}$ with $\mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \\ y_3 \end{bmatrix}$ and $\mathbf{d} = \begin{bmatrix} -16 \\ 0 \\ 7 \end{bmatrix}$, the value of $y_2$ is

A

29

B

27

C

25

D

5

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10

2026 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Mining Engineering (MN) 2026

A mine workshop needs to assign 4 jobs to 4 service engineers. The cost for performing a job by an individual service engineer is given. A typical job can be assigned to only one service engineer. If the service engineer S1 cannot perform the job J3, and S3 cannot perform the job J4, the optimal cost for completion of jobs is __________. (answer in integer)

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11

2026 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Naval Architecture and Marine Engineering (NM) 2026

The determinant of the matrix \(\begin{pmatrix} 1 & p & q+r \\ 1 & q & r+p \\ 1 & r & p+q \end{pmatrix}\) is ______.

A

pqr

B

p + q + r

C

pq + qr + rp

D

0

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12

2026 · Naval Architecture & Marine Engineering · Engineering Mathematics · Linear Algebra

Naval Architecture and Marine Engineering (NM) 2026

Which of the following statements is/are TRUE regarding matrices?

A

Every square matrix can be expressed as the sum of a symmetric and a skew-
symmetric matrix.

B

The determinant of a square matrix is same as that of its transpose.

C

The product of a square matrix and its inverse results in the identity matrix.

D

Eigenvalues of a square matrix are always real and positive.

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