System of Linear Equations Matrix Method MCQ

16. The following system of equations

x₁ + x₂ + 2x₃ = 1
x₁ + 2x₂ + 3x₃ = 2
x₁ + 4x₂ + ax₃ = 4

has a unique solution. The only possible value(s) for a is/are

1. 0
2. either 0 or 1
3. one of 0, 1 or -1
4. any real number other than 5

17. The system of linear equations

4x + 2y = 7
2x + y = 6

has

1. a unique solution
2. no solution
3. an infinite number of solutions
4. exactly two distinct solutions

18. For what value of a, if any, will the following system of equations in x, y and z have a solution?

2x + 3y = 4
x + y + z = 4
x + 2y – z = a

1. any real number
2. 0
3. 1
4. there is no such value

19. The following simultaneous equations

x + y + z = 3
x + 2y + 3z = 4
x + 4y + kz = 6

will not have a unique solution for k equal to

1. 0
2. 5
3. 6
4. 7

20. For what values of α and β, the following simultaneous equations have an infinite number of solutions?

x + y + z = 5
x + 3y + 3z = 9
x + 2y + αz = β

1. 2, 7
2. 3, 8
3. 8, 3
4. 7, 2

21. Solution for the system defined by the set of equations

4y + 3z = 8
2x – z = 2
3x + 2y = 5 is

1. x = 0; y = 1; z = 4/3
2. x = 0; y = 1/2; z = 2
3. x = 1; y = 1/2; z = 2
4. non-existent

22. Consider the following system of equations in three real variables x₁, x₂ and x₃

2x₁ – x₂ + 3x₃ = 1
3x₁ – 2x₂ + 5x₃ = 2
-x₁ – 4x₂ + x₃ = 3

This system of equations has

1. no solution
2. a unique solution
3. more than one but a finite number of solutions
4. an infinite number of solutions

23. Consider a non-homogenous system of linear equations representing mathematically an overdetermined system. Such a system will be

1. consistent having a unique solution
2. consistent having many solutions
3. inconsistent having a unique solution
4. inconsistent having no solution

24. How many solutions does the following system of linear equations have?

-x + 5y = -1
x – y = 2
x + 3y = 3

1. infinitely many
2. two distinct solutions
3. unique
4. none

25. Consider the following system of linear equations

\[\begin{bmatrix} 2 & 1 & -4\\ 4 & 3 & -12\\ 1 & 2 & -8 \end{bmatrix} \begin{bmatrix} x\\ y\\ z \end{bmatrix}=\begin{bmatrix} \alpha \\ 5\\ 7 \end{bmatrix}\]

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of α, does this system of equations have infinitely many solutions?

1. 0
2. 1
3. 2
4. infinitely many

26. Consider the system of simultaneous equations

x + 2y + z = 6
2x + y + 2z = 6
x + y + z = 5

This system has

1. unique solution
2. infinite number of solutions
3. no solution
4. exactly two solutions