List of top Data Science and Artificial Intelligence Questions on Linear Algebra

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

Consider a set \(S_1 = \{x = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \mid x^Tx \leq 16\}\). Let \(S_2\) be another set which is a subspace of \(\mathbb{R}^3\) with dimension two.
Which of the following gives the area of \(S_1 \cap S_2\)?

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 \(\gamma_1 + \gamma_2 + \gamma_3 = 1 + \sqrt{2}\)?

Let \(M = \left(I_n - \frac{1}{n}\mathbf{1}\mathbf{1}^T\right)\) be a matrix, where \(\mathbf{1} = (1,1,1,\ldots,1)^T \in \mathbb{R}^n\) and \(I_n\) is the identity matrix of order \(n\).

Which of the following options is/are correct?

Let \(A = \left(I_n - \frac{1}{n}\mathbf{1}\mathbf{1}^T\right)\) be a matrix, where \(\mathbf{1} = (1,1,1,\ldots,1)^T \in \mathbb{R}^n\) and \(I_n\) is the identity matrix of order \(n\).

The value of \(\max_{S} x^TAx\), where \(S = \{x \in \mathbb{R}^n \mid x^Tx = 1\}\), is __________. (Answer in integer)

Consider the directed acyclic graph (DAG) below :

Which of the following is/are valid vertex orderings that can be obtained from a topological sort of the DAG ?

Let \(\R\) be the set of real numbers, U be a subspace of \(\R^3\) and M ∈ \(\R^{3×3}\) be the matrix corresponding to the projection on to the subspace U.
Which of the following statements is/are TRUE ?

Which of the following statements is/are TRUE ?
Note: \(\R\) denotes the set of real numbers.

Select all choices that are subspaces of \(\R^3\).
Note : \(\R\) denotes the set of real numbers.

Consider the two neural networks (NNs) shown in Figures 1 and 2, with ReLU activation (ReLU(z) = max{0, z}, ∀z ∈ R). R denotes the set of real numbers. The connections and their corresponding weights are shown in the Figures. The biases at every neuron are set to 0. For what values of p, q, r in Figure 2 are the two NNs equivalent, when x₁, x₂, x₃ are positive ?