Let \(A\) be a \(10 \times 10\) matrix such that \(A^{5}\) is a null matrix, and let \(I\) be the \(10 \times 10\) identity matrix. The determinant of \(A + I\) is \(\underline{\hspace{1cm}}\).
Show Hint
Correct Answer: 1
Solution and Explanation
Nilpotent matrices have all eigenvalues equal to zero.
Thus eigenvalues of \(A + I\) are: \[ 1, 1, 1, \dots, 1 (10\ \text{times}) \] Hence, \[ \det(A+I) = 1^{10} = 1. \]
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 EE exam
A continuous time periodic signal \( x(t) \) is given by: \[ x(t) = 1 + 2\cos(2\pi t) + 2\cos(4\pi t) + 2\cos(6\pi t) \] If \( T \) is the period of \( x(t) \), then evaluate: \[ \frac{1}{T} \int_0^T |x(t)|^2 \, dt \quad {(round off to the nearest integer).} \]
- GATE EE - 2025
- Digital Signal Processing
The maximum percentage error in the equivalent resistance of two parallel connected resistors of 100 \( \Omega \) and 900 \( \Omega \), with each having a maximum 5% error, is: \[ {(round off to nearest integer value).} \]
- GATE EE - 2025
- Electrical Energy, Power
Consider a distribution feeder, with \( R/X \) ratio of 5. At the receiving end, a 350 kVA load is connected. The maximum voltage drop will occur from the sending end to the receiving end, when the power factor of the load is: \[ {(round off to three decimal places).} \]
In the circuit with ideal devices, the power MOSFET is operated with a duty cycle of 0.4 in a switching cycle with \( I = 10 \, {A} \) and \( V = 15 \, {V} \). The power delivered by the current source, in W, is: \[ {(round off to the nearest integer).} \]
- GATE EE - 2025
- Electrical Circuits
The induced emf in a 3.3 kV, 4-pole, 3-phase star-connected synchronous motor is considered to be equal and in phase with the terminal voltage under no-load condition. On application of a mechanical load, the induced emf phasor is deflected by an angle of \( 2^\circ \) mechanical with respect to the terminal voltage phasor. If the synchronous reactance is \( 2 \, \Omega \), and stator resistance is negligible, then the motor armature current magnitude, in amperes, during loaded condition is closest to: \[ {(round off to two decimal places).} \]
- GATE EE - 2025
- Electrical Energy, Power