Linear Algebra - Data Science & Artificial Intelligence Previous Year Questions

1

2024 · Data Science & Artificial Intelligence · Linear Algebra · Eigenvalues, Projections and Quadratic Forms

Data Science & Artificial Intelligence (DA) 2024

Consider the matrix \( M = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix} \).
Which ONE of the following statements is TRUE?

A

The eigenvalues of \(M \) are non-negative and real.

B

The eigenvalues of \(M \) are complex conjugate pairs.

C

One eigenvalue of \(M \) is positive and real, and another eigenvalue of \(M \) is zero.

D

One eigenvalue of \(M \) is non-negative and real, and another eigenvalue of \(M \) is negative and real.

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2

2025 · Data Science & Artificial Intelligence · Linear Algebra · Linear Systems and Matrix Decompositions

Data Science & Artificial Intelligence (DA) 2025

The number of additions and multiplications involved in performing Gaussian elimination on any \(n \times n\) upper triangular matrix is of the order

A

\(O(n)\)

B

\(O(n^2)\)

C

\(O(n^3)\)

D

\(O(n^4)\)

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3

2026 · Data Science & Artificial Intelligence · Linear Algebra · Eigenvalues, Projections and Quadratic Forms

Data Science and Artificial Intelligence (DA) 2026

For a classification problem, Principal Component Analysis (PCA) has been used to reduce the dimensionality of a feature space from 100 to 10.
Which of the following options is true about the angle \( \theta \) between the first and the tenth principal components?

A

\(\theta = 0^\circ \)

B

\(\theta = 90^\circ \)

C

\(90^\circ < \theta \le 180^\circ \)

D

\(0 < \theta < 90^\circ \)

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4

2024 · Data Science & Artificial Intelligence · Linear Algebra · Eigenvalues, Projections and Quadratic Forms

Data Science & Artificial Intelligence (DA) 2024

Consider the \(3 \times 3\) matrix \(M = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6 \end{bmatrix}\).
The determinant of \((M^2 + 12M)\) is ______.

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5

2025 · Data Science & Artificial Intelligence · Linear Algebra · Eigenvalues, Projections and Quadratic Forms

Data Science & Artificial Intelligence (DA) 2025

The sum of the elements in each row of \(A \in \mathbb{R}^{n \times n}\) is 1. If \(B = A^3 - 2A^2 + A\), which one of the following statements is correct (for \(x \in \mathbb{R}^n\))?

A

The equation \(Bx = 0\) has no solution

B

The equation \(Bx = 0\) has exactly two solutions

C

The equation \(Bx = 0\) has infinitely many solutions

D

The equation \(Bx = 0\) has a unique solution

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6

2026 · Data Science & Artificial Intelligence · Linear Algebra · Eigenvalues, Projections and Quadratic Forms

Data Science and Artificial Intelligence (DA) 2026

Let \( M = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \) be a \( 2 \times 2 \) matrix, where \( \theta = \frac{2\pi}{5} \), and \( I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \).
Which of the following options is equal to \( M^{2026} \)?

A

\(M^2 \)

B

\(M \)

C

\(M^{-1} \)

D

\(I_2 \)

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