Eigenvalues, Projections and Quadratic Forms - Linear Algebra - Data Science & Artificial Intelligence Previous Year Questions

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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

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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3

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

Data Science & Artificial Intelligence (DA) 2024

Let \(\mathbf{u} = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{bmatrix}\), and let \(\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5\) be the singular values of the matrix \(\mathbf{M} = \mathbf{u}\mathbf{u}^T\) (where \(\mathbf{u}^T\) is the transpose of \(\mathbf{u}\)). The value of \(\sum_{i=1}^{5} \sigma_i\) is ______.

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4

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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5

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

Data Science & Artificial Intelligence (DA) 2025

Let \(A = I_n + x x^\top\), where \(I_n\) is the \(n \times n\) identity matrix and \(x \in \mathbb{R}^n, x^\top x = 1\). Which of the following options is/are correct?

A

Rank of \(A\) is \(n\)

B

\(A\) is invertible

C

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

D

\(A^{-1}\) has a negative eigenvalue

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6

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

Data Science & Artificial Intelligence (DA) 2025

Let \(x_1, x_2, x_3, x_4, x_5\) be a system of orthonormal vectors in \(\mathbb{R}^{10}\). Consider the matrix \(A = x_1x_1^T + \dots + x_5x_5^T\). Which of the following statements is/are correct?

A

Singular values of \(A\) are also its eigenvalues

B

Singular values of \(A\) are either 0 or 1

C

Determinant of \(A\) is 1

D

\(A\) is invertible

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7

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

Data Science & Artificial Intelligence (DA) 2025

Let \(D = \{x^{(1)}, \ldots, x^{(n)}\}\) be a dataset of \(n\) observations where each \(x^{(i)} \in \mathbb{R}^{100}\). It is given that \(\sum_{i=1}^n x^{(i)} = 0\). The covariance matrix computed from \(D\) has eigenvalues \(\lambda_i = 100^{2-i}\), \(1 \leq i \leq 100\). Let \(u \in \mathbb{R}^{100}\) be the direction of maximum variance with \(u^T u = 1\). The value of \(\frac{1}{n} \sum_{i=1}^n (u^T x^{(i)})^2 = \)__________ (Answer in integer)

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8

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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9

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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10

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

Data Science and Artificial Intelligence (DA) 2026

Let \(\gamma_1, \gamma_2, \gamma_3\) be the eigenvalues of the matrix \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & -\sin t & \cos t \end{bmatrix}\), where \(t \in [-\pi, \pi]\) is in radians.
Which one of the following options lists all the possible values of \(t\) satisfying \(\gamma_1 + \gamma_2 + \gamma_3 = 1 + \sqrt{2}\)?

A

\(\{\frac{\pi}{3}, -\frac{\pi}{4}\}\)

B

\(\{\frac{\pi}{4}, -\frac{\pi}{3}\}\)

C

\(\{\frac{\pi}{4}, -\frac{\pi}{4}\}\)

D

\(\{\frac{\pi}{3}, -\frac{\pi}{3}\}\)

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11

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

Data Science and Artificial Intelligence (DA) 2026

Let A = \(\left(I_n - \frac{1}{n} \mathbf{1}\mathbf{1}^T\right)\) be a matrix, where \(\mathbf{1} = (1,1,1,...,1)^T \in \mathbb{R}^n\) and In is the identity matrix of order n.
The value of \(\max_{x} x^T A x\), where S = {x ∈ ℝⁿ | x^T x = 1}, is __________ . (Answer in integer)

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