List of top Statistics Questions on Matrix algebra asked in IIT JAM MS

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

Let a unit vector \( \mathbf{v} = (v_1 \, v_2 \, v_3)^T \) be such that \( A\mathbf{v} = 0 \), where

\[ A = \begin{pmatrix} \frac{5}{6} & -\frac{1}{3} & -\frac{1}{6} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ -\frac{1}{6} & \frac{1}{3} & \frac{5}{6} \end{pmatrix}. \]

Then the value of \( \sqrt{6} (|v_1| + |v_2| + |v_3|) \) equals

Let \( u, v \in \mathbb{R}^4 \) be such that

\[ u = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 5 \end{pmatrix} \quad \text{and} \quad v = \begin{pmatrix} 5 \\ 3 \\ 2 \\ 1 \end{pmatrix} \]

Then the equation \( u^T x = v \) has

For \( j = 1, 2, \dots, 5 \), let \( P_j \) be the matrix of order \( 5 \times 5 \) obtained by replacing the \( j^{th} \) column of the identity matrix of order \( 5 \times 5 \) with the column vector \( v = (5, 4, 3, 2, 1)^T \). Then the determinant of the matrix product \( P_1 P_2 P_3 P_4 P_5 \) is

Let \( Q, A, B \) be matrices of order \( n \times n \) with real entries such that \( Q \) is orthogonal and \( A \) is invertible. Then the eigenvalues of \( Q^T A^{-1} B Q \) are always the same as those of

Let \( \mathbf{v} \in \mathbb{R}^k \) with \( \mathbf{v}^T \mathbf{v} \neq 0 \). Let \[ P = I - 2 \frac{ \mathbf{v} \mathbf{v}^T }{ \mathbf{v}^T \mathbf{v} }, \] where \( I \) is the \( k \times k \) identity matrix. Then which of the following statements is (are) TRUE?

Let a unit vector \( \mathbf{v} = (v_1, v_2, v_3)^T \) be such that \( A\mathbf{v} = 0 \), where
\[ A = \begin{pmatrix} \frac{5}{6} & \frac{1}{3} & -\frac{1}{6} \\ \frac{1}{3} & \frac{1}{3} & -\frac{1}{3} \\ -\frac{1}{6} & \frac{1}{3} & \frac{5}{6} \end{pmatrix}. \] Then the value of \( \sqrt{6} \left( |v_1| + |v_2| + |v_3| \right) \) equals

For \( j = 1, 2, \dots, 5 \), let \( P_j \) be the matrix of order \( 5 \times 5 \) obtained by replacing the \( j^{th} \) column of the identity matrix of order \( 5 \times 5 \) with the column vector \( v = (5, 4, 3, 2, 1)^T \). Then the determinant of the matrix product \( P_1 P_2 P_3 P_4 P_5 \) is

Let \( \mathbf{v} \in \mathbb{R}^k \) with \( \mathbf{v}^T \mathbf{v} \neq 0 \). Let \[ P = I - 2 \frac{ \mathbf{v} \mathbf{v}^T }{ \mathbf{v}^T \mathbf{v} }, \] where \( I \) is the \( k \times k \) identity matrix. Then which of the following statements is (are) TRUE?

Let \( Q, A, B \) be matrices of order \( n \times n \) with real entries such that \( Q \) is orthogonal and \( A \) is invertible. Then the eigenvalues of \( Q^T A^{-1} B Q \) are always the same as those of