List of top Statistics Questions on Matrix Operations

Let \[ P = \begin{pmatrix} 0 & 1 & 1 & 1 & 1 \\ -1 & 0 & 1 & 1 & 1 \\ -1 & -1 & 0 & 1 & 1 \\ -1 & -1 & -1 & 0 & 1 \\ -1 & -1 & -1 & -1 & 0 \end{pmatrix} \] If \( \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5 \) are eigenvalues of \( P \), then \( \prod_{i=1}^{5} \lambda_i = \) ________ (answer in integer).

Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear map defined by \[ T(x_1, x_2, x_3) = (3x_1 + 5x_2 + x_3, x_3, 2x_1 + 2x_3). \] {Then the rank of \( T \) is equal to _______ (answer in integer).}

Let \[ P = \begin{pmatrix} 0 & 1 & 1 & 1 & 1 \\ -1 & 0 & 1 & 1 & 1 \\ -1 & -1 & 0 & 1 & 1 \\ -1 & -1 & -1 & 0 & 1 \\ -1 & -1 & -1 & -1 & 0 \end{pmatrix} \] If \( \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5 \) are eigenvalues of \( P \), then \( \prod_{i=1}^{5} \lambda_i = \) __________ (answer in integer).

Let \( \mathcal{O} = \{ P : P \text{ is a } 3 \times 3 \text{ real matrix satisfying } P^T P = I_3 \text{ and } \det(P) = 1 \}, \)
where \( I_3 \) denotes the identity matrix of order 3. Then which of the following options is/are correct?

Let \( P = (a_{ij}) \) be a \( 10 \times 10 \) matrix with \[ a_{ij} = \begin{cases} -\frac{1}{10} & \text{if } i \neq j, \\ \frac{9}{10} & \text{if } i = j. \end{cases} \] Then the rank of \( P \) equals:

Let \( \{X_n\}_{n \geq 1} \) be a Markov chain with state space \( \{1, 2, 3\} \) and transition probability matrix \[ P = \begin{bmatrix} \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \] Then \( P(X_2 = 1 \mid X_1 = 1, X_3 = 2) \) (rounded off to two decimal places) equals:

Suppose that \( (X_1, X_2, X_3) \) has a \( N_3(\mu, \Sigma) \) distribution with \[ \mu = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \quad \text{and} \quad \Sigma = \begin{pmatrix} 2 & 2 & 1 \\ 2 & 5 & 1 \\ 1 & 1 & 1 \end{pmatrix}. \] Given that \( \Phi(-0.5) = 0.3085 \), where \( \Phi(.) \) denotes the cumulative distribution function of a standard normal random variable, \[ P\left( (X_1 - 2X_2 + 2X_3)^2 < \frac{7}{2} \right) \text{ (rounded off to two decimal places) equals ...............} \]

Let \( A \) be a \( 3 \times 3 \) real matrix such that \[ A \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ 0 \end{bmatrix}, \quad A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ 0 \end{bmatrix}, \quad A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 4 \end{bmatrix}. \] Then which of the following statements is/are true?

Consider the orthonormal set \[ v_1 = \begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}, v_2 = \begin{bmatrix} \frac{1}{\sqrt{6}} \\ \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} \end{bmatrix}, v_3 = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ \frac{1}{\sqrt{2}} \end{bmatrix} \] with respect to the standard inner product on \( \mathbb{R}^3 \). If \( u = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \) is the vector such that inner products of \( u \) with \( v_1, v_2 \) and \( v_3 \) are 1, 2 and 3, respectively, then \( a^2 + b^2 + c^2 \) (in integer) equals ................

Consider the following statements. (I) Let \( A \) and \( B \) be two \( n \times n \) real matrices. If \( B \) is invertible, then \( \text{rank}(BA) = \text{rank}(A) \). (II) Let \( A \) be an \( n \times n \) real matrix. If \( A^2x = b \) has a solution for every \( b \in \mathbb{R}^n \), then \( Ax = b \) also has a solution for every \( b \in \mathbb{R}^n \). Which of the above statements is/are true?

Let \( A \) be an \( n \times n \) real matrix. Consider the following statements.
(I) If \( A \) is symmetric, then there exists \( c \geq 0 \) such that \( A + c I_n \) is symmetric and positive definite, where \( I_n \) is the \( n \times n \) identity matrix.
(II) If \( A \) is symmetric and positive definite, then there exists a symmetric and positive definite matrix \( B \) such that \( A = B^2 \).
\text{Which of the above statements is/are true?}

Let \( A \) be a \( 2 \times 2 \) real matrix such that \( AB = BA \) for all \( 2 \times 2 \) real matrices \( B \). If the trace of \( A \) equals 5, then the determinant of \( A \) (rounded off to two decimal places) equals ................

Let \( A \) be a \( 3 \times 3 \) real matrix having eigenvalues \( 1, 0, \) and \( -1 \). If \( B = A^2 + 2A + I_3 \), where \( I_3 \) is the \( 3 \times 3 \) identity matrix, then which one of the following statements is true?