System of Linear Equations Matrix Method MCQ

1. The solution of the system of equations

x + y + z = 4
x – y + z = 0
2x + y + z = 5 is

1. x = 2, y = 2, z = 0
2. x = 1, y = 4, z = 1
3. x = 2, y = 4, z = 3
4. x = 1, y = 2, z = 1

2. Consider the systems, each consisting of m inear equations in n variables.

1. if m < n, then all such systems have a solution.
2. if m > n, then none of these systems has a solution.
3. if m = n, then there exists a system which has a solution

Which one of the following is correct?

1. i, ii and iii are true
2. only ii and iii are true
3. only iii is true
4. none of them is true

3. The solution to the system of equations is

\[\begin{bmatrix} 2 &5 \\ -4 &3 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}=\begin{bmatrix} 2\\ -30 \end{bmatrix}\]

1. 6, 2
2. -6, 2
3. -6, -2
4. 6, -2

4. Consider the following linear system.

x + 2y – 3z = a
2x + 3y + 3z = b
5x + 9y – 6z = c

This system is consistent if a, b and c satisfy the equation

1. 7a – b – c = 0
2. 3a + b – c = 0
3. 3a – b + c = 0
4. 7a – b + c = 0

5. Let A be an n x n matrix with rank r (o < r < n). Then AX = 0 has p independent solutions, where p is

1. r
2. n
3. n – r
4. n + r

6. If the following system has non-trivial solution,

px + qy + rz = 0
qx + ry + pz = 0
rx + py + qz = 0

1. p – q + r = 0 or p = q = -r
2. p + q – r = 0 or p = -q = r
3. p + q + r = 0 or p = q = r
4. p – q + r = 0 or p = -q = -r

7. Consider a system of linear equations:

x – 2y + 3z = -1
x – 3y + 4z = 1
-2x + 4y – 6z = k

The value of k for which the system has infinitely many solution is

1. 1
2. 2
3. 3
4. 4

8. For what value of p the following set of equations will have no solution?

2x + 3y = 5
3x + py = 10

1. 1
2. 2.5
3. 3.5
4. 4.5

9. Consider the following system of equations

3x + 2y = 1
4x + 7z = 1
x + y + z =3
x – 2y + 7z = 0

The number of solutions for this system is

1. 1
2. 2
3. 3
4. 4

10. The system of linear equations

\[\begin{bmatrix} 2 & 1 & 3\\ 3 & 0 & 1\\ 1 & 2 & 5 \end{bmatrix} \begin{bmatrix} a\\ b\\ c \end{bmatrix}=\begin{bmatrix} 5\\ -4\\ 14 \end{bmatrix}\]

1. a unique solution
2. infinitely many solutions
3. no solution
4. exactly two solutions

11. Given a system of equations:

x + 2y + 2z = b₁
5x + y + 3z = b₂

Which of the following is true regarding its solution?

1. The system has a unique solution for any given b₁ and b₂
2. The system will have infinitely many solutions for any given b₁ and b₂
3. Whether or not a solution exists depends on the given b₁ and b₂
4. The system would have no solution for any values of b₁ and b₂

12. 

x + 2y + z = 4
2x + y + 2z = 5
x – y + z = 1

The system of algebraic given below has

1. a unique solution of x = 1, y = 1 and z = 1
2. only the two solutions of ( x = 1, y = 1,  z = 1) and ( x = 2, y = 1,  z = 0)
3. infinite number of solutions
4. no feasible solution

13. The system of equations

x + y + z = 6
x + 4y + 6z = 20
x + 4y + λz = μ

has no solution for values of λ and μ given by 

1. λ = 6, μ = 20
2. λ = 6, μ ≠ 20
   
3. λ ≠ 6, μ = 20
   
4. λ ≠ 6, μ ≠ 20
   

14. Consider the following system of equations

2x₁ + x₂ + x₃ = 0
x₂ – x₃ = 0
x₁ + x₂ = 0

This system has

1. a unique solution
2. no solution
3. infinite number of solutions
4. five solutions

15. For the set of equations

x₁ + 2x₂ + x₃ + 4x₄ = 2
3x₁ + 6x₂ + 3x₃ + 12x₄ = 6

the following statement is true

1. only the trivial solution x₁ = x₂ = x₃ = x₄ = 0 exists
2. there is no solution
3. a unique non-trivial solution exists
4. multiple non-trivial solutions exists