List of practice Questions

Let $ D = \{ x^{(1)}, x^{(2)}, \dots, 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 $, for $ 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

\[ \frac{1}{n} \sum_{i=1}^{n} \left( u^T x^{(i)} \right)^2 = \underline{\hspace{2cm}} \quad \text{(Answer in integer)} \]

Let $ D = \{ x^{(1)}, x^{(2)}, \dots, 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 $, for $ 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 \[ \frac{1}{n} \sum_{i=1}^{n} \left( u^T x^{(i)} \right)^2 = \underline{\hspace{2cm}} \quad \text{(Answer in integer)} \]

Consider a linear homogeneous system of equations $ Ax = 0 $, where $ A $ is an $ n \times n $ matrix, $ x $ is an $ n \times 1 $ vector, and $ 0 $ is an $ n \times 1 $ null vector. Let $ r $ be the rank of $ A $. For a non-trivial solution to exist, which of the following conditions is/are satisfied?

Given matrix $ A = \left[\begin{matrix} x & 1 & 3 \\ y & 2 & 6 \\ 3 & 5 & 7 \end{matrix}\right] $, the ordered pair $ (x, y) $ for which $ \text{det}(A) = 0 $ is

If $ A = \begin{bmatrix} 10 & 2k + 5 \\ 3k - 3 & k + 5 \end{bmatrix} $ is a symmetric matrix, the value of $ k $ is ________________.

Suppose that the characteristic equation of $ M \in \mathbb{C}^{3 \times 3} $ is \[ \lambda^3 + \alpha \lambda^2 + \beta \lambda - 1 = 0, \] where $ \alpha, \beta \in \mathbb{C} $ with $ \alpha + \beta \neq 0 $.
Which of the following statements is TRUE?

$A$ is a $3\times 5$ real matrix of rank $2$. For the set of homogeneous equations $A x = 0$, where $0$ is a zero vector and $x$ is a vector of unknown variables, which of the following is/are true?

A, B, C and D are vectors of length 4. The rank of the matrix
A = $\begin{bmatrix} a_1 & a_2 & a_3 & a_4 \end{bmatrix}$,
B = $\begin{bmatrix} b_1 & b_2 & b_3 & b_4 \end{bmatrix}$,
C = $\begin{bmatrix} c_1 & c_2 & c_3 & c_4 \end{bmatrix}$,
D = $\begin{bmatrix} d_1 & d_2 & d_3 & d_4 \end{bmatrix}$
It is known that B is not a scalar multiple of A. Also, C is linearly independent of A and B. Further, $ D = 3A + 2B + C $

The rank of the matrix $ \begin{bmatrix} a_1 & a_2 & a_3 & a_4 \\ b_1 & b_2 & b_3 & b_4 \\ c_1 & c_2 & c_3 & c_4 \\ d_1 & d_2 & d_3 & d_4 \end{bmatrix} $ is $\underline{\hspace{1cm}}$

Let $A$ be an $n \times n$ square matrix ($n>1$) with elements \[ a_{ij} = \begin{cases} i \times j, & \text{if } i \ge j \\ 0, & \text{if } i < j \end{cases} \] The determinant of $A$ is:

For the matrix \[ \begin{bmatrix} 3 & 1 & 2 \\ 2 & -3 & -1 \\ 1 & 2 & 1 \end{bmatrix} \] find the ratio of the product of eigenvalues to the sum of eigenvalues (round off to nearest integer).

Find the rank of the matrix: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 2 \\ 2 & 5 & 6 & 4 \\ 2 & 6 & 8 & 5 \end{bmatrix} \] Rank = ?

For real constants $a$ and $b$, let \[ M = \begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} \\ a & b \end{bmatrix} \] be an orthogonal matrix. Then which of the following statements is/are always TRUE?

Let $M$ be an $n \times n$ non-zero skew symmetric matrix. Then the matrix $(I_n - M)(I_n + M)^{-1}$ is always

Let $T: \mathbb{R}^3 \to \mathbb{R}^4$ be a linear transformation. If $T(1,1,0) = (2,0,0,0)$, $T(1,0,1) = (2,4,0,0)$, and $T(0,1,1) = (0,0,2,0)$, then $T(1,1,1)$ equals

If

\[ P = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 0 & 1 \\ 0 & 0 & -1 \end{bmatrix} \quad \text{and} \quad 6P^{-1} = aI_3 + bP - P^2, \quad \text{then the ordered pair} \quad (a,b) \quad \text{is} \]

Let $ P $ be a $ 2 \times 2 $ real matrix such that every non-zero vector in $ \mathbb{R}^2 $ is an eigenvector of $ P $. Suppose that $ \lambda_1 $ and $ \lambda_2 $ denote the eigenvalues of $ P $ and $ P \left[ \begin{array}{c} \sqrt{2} \\ \sqrt{3} \end{array} \right] = \left[ \begin{array}{c} 2 \\ t \end{array} \right] $ for some $ t \in \mathbb{R} $. Which of the following statements is (are) TRUE?

If

\[ \begin{pmatrix} \frac{\sqrt{5}}{3} & -\frac{2}{3} & c \\ \frac{2}{3} & \frac{\sqrt{5}}{3} & d \\ a & b & 1 \end{pmatrix} \]

is a real orthogonal matrix, then $ a^2 + b^2 + c^2 + d^2 $ equals ...............

If $ ad - bc = 2 $ and $ ps - qr = 1 $, then the determinant of

\[ \begin{pmatrix} a & b & 0 & 0 \\ 3 & 10 & 2p & q \\ c & d & 0 & 0 \\ 2 & 7 & 2r & s \end{pmatrix} \]

equals ............

Let $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ be a linear transformation such that $ T\left( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 0 \end{bmatrix} $ and $ T\left( \begin{bmatrix} 3 \\ 2 \end{bmatrix} \right) = \begin{bmatrix} a \\ b \end{bmatrix} $. Then $ \alpha + \beta + a + b $ equals

Let $ P $ be an $ n \times n $ non-null real skew-symmetric matrix, where $ n $ is even. Which of the following statements is (are) always TRUE?

Let $ P $ be a $ 3 \times 3 $ non-null real matrix. If there exists a $ 3 \times 2 $ real matrix $ Q $ and a $ 2 \times 3 $ real matrix $ R $ such that $ P = QR $, then