Consider a Leniar Homogeneous system AX = 0, where A is N × N matrix and X is N × 1 vector. When we solve this 0 is null matrix of N × 1. Let R be the rank matrix for a for a Non-trivial solution.
\(R = N\)
\(|A| ≠ 0\)
\(|A| = 0\)
\(R < N\)
The Correct Option is C, D
Solution and Explanation
The correct answer is \(|A| = 0\) and \(R < N\)
Top Questions on Matrix Algebra
For the matrix [A] given below, the transpose is __________.
\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{bmatrix} \]- GATE CE - 2025
- Mathematics
- Matrix Algebra
Let the matrix $ A = \begin{pmatrix} 1 & 0 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \end{pmatrix} $ satisfy $ A^n = A^{n-2} + A^2 - I $ for $ n \geq 3 $. Then the sum of all the elements of $ A^{50} $ is:
- JEE Main - 2025
- Mathematics
- Matrix Algebra
Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \]Then \( n(S) \) is equal to ______.
- JEE Main - 2025
- Mathematics
- Matrix Algebra
Let \( A \) be a \( 3 \times 3 \) real matrix such that \[ A^{2}(A - 2I) - 4(A - I) = O, \] where \( I \) and \( O \) are the identity and null matrices, respectively.
If \[ A^{5} = \alpha A^{2} + \beta A + \gamma I, \] where \( \alpha, \beta, \gamma \) are real constants, then \( \alpha + \beta + \gamma \) is equal to:- JEE Main - 2025
- Mathematics
- Matrix Algebra
Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \]Then \( n(S) \) is equal to ______.
- JEE Main - 2025
- Mathematics
- Matrix Algebra
Questions Asked in GATE CH exam
An ideal monoatomic gas is contained inside a cylinder-piston assembly connected to a Hookean spring as shown in the figure. The piston is frictionless and massless. The spring constant is 10 kN/m. At the initial equilibrium state (shown in the figure), the spring is unstretched. The gas is expanded reversibly by adding 362.5 J of heat. At the final equilibrium state, the piston presses against the stoppers. Neglecting the heat loss to the surroundings, the final equilibrium temperature of the gas is __________ K (rounded off to the nearest integer).
- GATE CH - 2025
- Thermodynamics
- Consider the matrix: \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \] The eigenvalues of the matrix are 0.27 and __________ (rounded off to 2 decimal places).
- GATE CH - 2025
- Linear Algebra
The residence-time distribution (RTD) function of a reactor (in min$^{-1}$) is
The mean residence time of the reactor is __________ min (rounded off to 2 decimal places).}- GATE CH - 2025
- Chemical Equations
Ideal nonreacting gases A and B are contained inside a perfectly insulated chamber, separated by a thin partition, as shown in the figure. The partition is removed, and the two gases mix till final equilibrium is reached. The change in total entropy for the process is _________J/K (rounded off to 1 decimal place).
Given: Universal gas constant \( R = 8.314 \) J/(mol K), \( T_A = T_B = 273 \) K, \( P_A = P_B = 1 \) atm, \( V_B = 22.4 \) L, \( V_A = 3V_B \).- GATE CH - 2025
- Thermodynamics
The following data is given for a ternary \(ABC\) gas mixture at 12 MPa and 308 K:
\(y_i\): mole fraction of component \(i\) in the gas mixture
\(\hat{\phi}_i\): fugacity coefficient of component \(i\) in the gas mixture at 12 MPa and 308 K
The fugacity of the gas mixture is __________ MPa (rounded off to 3 decimal places).- GATE CH - 2025
- Thermodynamics