1. Let the matrix A be the idempotent matrix then the Eigen values of the idempotent matrix are ________
a) 0, 1
b) 0
c) 1
d) -1
Explanation: According to the property of the Eigen values, the Eigen values of the idempotent matrix are either zero or unity.
So, the answer is 0 or 1.
2. Let us consider a 3×3 matrix A with Eigen values of λ1, λ2, λ3 and the Eigen values of A-1 are?
a) λ1, λ2, λ3
b) \( \frac{1}{λ_1}, \frac{1}{λ_2}, \frac{1}{λ_3}\)
c) -λ1, -λ2, -λ3
d) λ1, 0, 0
Explanation:According to the property of the Eigen values, if is the Eigen value of A, then \( \frac{1}{λ}\) is the Eigen value of A-1.
So the Eigen values of A-1 are \( \frac{1}{λ_1}, \frac{1}{λ_2}, \frac{1}{λ_3}\).
3. The Eigen values of a 3×3 matrix are λ1, λ2, λ3 then the Eigen values of a matrix A3 are __________
a) λ1, λ2, λ3
b) \( \frac{1}{λ_1}, \frac{1}{λ_2}, \frac{1}{λ_3}\)
c) \(λ_1^3, λ_2^3, λ_3^3\)
d) 1, 1, 1
Explanation: If λ1, λ2, λ3……… λn are the Eigen values of matrix A then the Eigen values of matrix Am are said to be \(λ_1^m, λ_2^m, λ_3^m,………λ_n^m\).
So, the answer is \(λ_1^3, λ_2^3, λ_3^3\).
4. Find the Eigen values of matrix A=\(\begin{bmatrix}
4 & 1 \\
1 & 4 \\
\end{bmatrix} \).
a) 3, -3
b) -3, -5
c) 3, 5
d) 5, 0
Explanation: According to the property of the Eigen value, the eigen values are determined as follows:
4 + 4 = 8
3 + 5 = 8
The sum of the Eigen values is equal to the sum of the principal diagonal elements of the matrix.
5. Where do we use Eigen values?
a) Fashion or cosmetics
b) Communication systems
c) Operations
d) Natural herbals
Explanation: Eigen values are used in communication systems, designing bridges, designing car stereo system, electrical engineering, mechanical companies
6. What are periodic signals?
a) The signals which change with time
b) The signals which change with frequency
c) The signal that repeats itself in time
d) The signals that repeat itself over a fixed frequency
Explanation: Those signals which repeat themselves in a fixed interval of time are called periodic signals. The continuous-time signal x(t) is periodic if and only if
x(t+T)= x(t).
7. Periodic signals are different in case of continuous time and discrete time signals.
a) True
b) False
Explanation: Periodic signals are same in case of continuous time and discrete time signals.
In case of continuous time signal, x(t)=x(t+T), for all t>0
In case of discrete time signal,
x(n)=x(n+N), for all n>0.
8. What is the time period of a periodic signal in actual terms?
a) The signals which start at t=-∞ and end at t=+∞
b) The signals which have a finite interval of occurrence
c) The signals which start at t= -∞ and ends at a finite time period
d) The signals which have a short period of occurrence
Explanation: The periodic signals have actually a time period between t=-∞ and at t= + ∞. These signals have an infinite time period, that is periodic signals are actually continued forever. But this is not possible in case of real time signals.
9. Periodic signals actually exist according to a definition.
a) True
b) False
Explanation: Periodic signals are defined as signals having time period in between t=-∞ and t=+ ∞. These signals have an infinite time period that is periodic signals are continued forever. But real time signals always cease at some time due to distortion and resistance.
10. What is a fundamental period?
a) Every interval of a periodic signal
b) Every interval of an aperiodic signal
c) The first interval of a periodic signal
d) The last interval of a periodic signal
Explanation: The first time interval of a periodic signal after which it repeats itself is called a fundamental period. It should be noted that the fundamental period is the first positive value of frequency for which the signal repeats itself.