Let \(M = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\) be a \(2 \times 2\) matrix, where \(…

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 subsp…

Let \(\gamma_1, \gamma_2, \gamma_3\) be the eigenvalues of the matrix \[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & -…

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{…

The number of additions and multiplications involved in performing Gaussian elimination on any \( n \times n \) upper triangular m…

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 fol…