MATLAB Questions and Answers – Linear Systems

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This set of MATLAB Multiple Choice Questions & Answers (MCQs) focuses on “Linear Systems”.

1. What is a linear system?
a) A system which follows homogeneity and additivity
b) A system which follows additivity
c) A system which follows homogeneity
d) Almost every system is linear

Explanation: A system which follows homogeneity and additivity is called linear system. This comes from the definition of superposition and here is the proof:
Taking a system with which produces y1(t) as output for an input x1(t) and an output y2(t) for an input x2(t),
HOMOGENITY: For input ax1(t), output should be ay1(t) and for input bx2(t), output should be by2(t).
ADDITIVITY: For a input of sum of x1(t) and x2(t), output should be the sum of a y1(t) and y2(t), i.e. the sum of individual response
Finally, if for an input of the sum of ax1(t) and bx2(t), if we get the output as sum of ay1(t) and by2(t) the system is both homogeneous and additive. This is similar to the superposition principle. No the system is linear.

2. What is the output if the following code?

```if(eig(A)==eig(A’))
disp(‘True’)```

a) True
b) No output
c) False
d) Error

Explanation: The syntax of the if control structure is wrong. While writing the logical expression, we cannot put it under brackets. If the brackets were removed, True would have been displayed.
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3. A student has to find a solution for a system of equations having three variables. He has defined the coefficient matrix as C while the variable matrix as d. He observes that the system is homogeneous. So, to find the solution, he must first check
a) Consistency
b) Homogeneity
c) Heterogeneity
d) Linearity

Explanation: If the system of equations is not consistent, the student cannot ever get a solution for the system of equation. He has to check consistency by finding the rank of the matrix. Thereafter he can comment on the nature of solutions the system possesses.

4. The command to find the eigen vector of a matrix in matrix form is _____________
a) eig(a,matrix)
b) eig(a,’matrix’)
c) eigen(a,matr)
d) eig(a)

Explanation: The in-built function to find the eigen values of any matrix in MATLAB is eig(a). But, to display the result in matrix form- we use eig(a,’matrix’). This nature of the eig command is in-built in MATLAB.

5. The nature of the eigen matrix displayed in MATLAB is ___________
a) Unsorted
b) Sorted
c) Sorted in Ascending order
d) Sorted in Descending order

Explanation: The eig(a,’matrix’) command will return the eigen values of the matrix ‘a’ in a matrix form. But, it will give it in an unsorted manner- the elements will be placed on the principal diagonal. To sort them, we need to use the sort command- this will sort the columns in ascending order.

6. Checking the linearity of a system, the first thing we need to check is whether the system is __________
a) Homogeneous or not
b) Consistent or not
c) Superposition
d) Depends on the representation of the system

Explanation: First we need to check if the system is represented by a set of equations or it is represented in terms of signals. If it is represented by a set of equation, we need to go for homogeneity first. If it is represented by the observation of input-output signals, we need to go by Superposition.

7. In MATLAB, how can we check linearity of systems represented by transfer function?
a) Check waveforms after superpostion
b) Compare rank of matrices
c) Go for Rouche’s Theorem
d) Intuition

Explanation: Since we have the transfer function of a system, we need to apply superposition and compare the waveforms generated after applying the method of superposition. Rouche’s Theorem is for a system of equations represented by matrices and it is the method of comparing ranks.

8. How can we check in MATLAB if an electrical circuit is linear or not?
a) Check consistency
b) Superposition
d) Check homogeneity

Explanation: We can model our circuit in Simulink and then apply the superposition theorem to check if at all the circuit is following superposition theorem. To apply the theorem, find the I/V relationship across any branch due to a single source and switch of the rest of the sources. Repeat this process for all the individual sources. Finally, turn on all the sources and check whether the I/V relationship is equal to the superposition of all the I/V sources previously calculated.

9. How can we find the solution of a nonhomogeneous system of equations without dealing with the rank of matrices?
a) Rouche’s theorem
b) Cramer’s rule
c) Gauss’s law
d) Cannot be done

Explanation: It is easier to find the solution of a system of equations for a non-homogeneous system using Cramer’s rule. Now, the process is time consuming since we need to find higher order determinants for higher order systems. This is why we go for Rouche’s theorem by hand. But MATLAB computes determinants very fast. Hence, without finding the rank, we can find the solution of a system of nonhomogeneous equations.

10. For a homogeneous system, Cramer’s rule will always yield a trivial solution in MATLAB.
a) True
b) False

Explanation: The process of Cramer’s rule will always yield a solution for the variables in a system. But for a homogeneous system with no constant value, it will yield a 0 as a solution. Hence all the variables will get a 0 as a solution, there-by yielding a trivial solution. This doesn’t imply that the system is inconsistent- this is why we go for comparing ranks, it will allow us to establish a more pertinent justification of the nature of the system.

11. To apply Cramer’s rule, the condition is _________
a) Non-homogeneous system of equations
b) Homogeneous system of equations
c) Determinant of co-efficient matrix is not equal to 0
d) No condition

Explanation: If the determinant of co-efficient matrix is 0, all the solutions will come to be Infinite. Thus, this is the condition to check before going for Cramer’s rule to check the solution of any system of equations.

12. To apply superposition in MATLAB, the condition, one of the condition is ___________
a) No active element is present except sources
b) No passive element is present
c) System should have unilateral elements
d) Nothing needs to be checked

Explanation: Superposition will definitely fail in presence of active elements. A system containing only passive elements will always follow superposition. Superposition theorem fails for unilateral elements., the network must have only passive elements.

13. A second order system with no initial condition is always linear.
a) True
b) False

Explanation: The superposition theorem will yield the nature of linearity of a system. For a system defined by an n-th order differential equation, if there are no initial conditions- the system will always be linear.

14. To check the rank of a non-singular square matrix, we have to use ___________
a) not required
b) rankm()
c) rankmatr()
d) rank()

Explanation: Since the matrix is non-singular, the rank is the highest number of rows or columns (both are the same for a square matrix. Since we know the matrix is non-singular, we don’t have to use any command. We can directly say the rank is equal to the order of the matrix. If the matrix is not singular, we will have to use rank() to get a value of the rank.

15. A student aims to use Cramer’s rule to find a solution for a homogeneous system. But while finding the solution, he observes that he is getting infinity as a solution. The code is

`p=X.\C`

X is the matrix created by replacing a column with constant co-efficients in the equation.
C is the co-efficient matrix
P is one of the variables in the system
Is this a proper justification?
a) Yes
b) No
c) Maybe
d) It is not possible to find solutions in MATLAB

Explanation: The code should have been p=X./C. Since the system is homogeneous, X is always 0. So, applying Cramer’s rule will always yield a trivial solution. But here the student gets or ‘Inf’ as an answer since according to his code, C will be divided by 0- this isn’t really the case. Hence, this is not a proper justification.

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