If x + 2y - 2u= 0, 2x - y - u = 0, x + 2z - u = 0, 4x -y + 3 z - u= 0 is a system of equations, then it isa)consistent with trivial solutionb)consistent without trivial solutionc)inconsistent with trivial solutiond)inconsistent without trivial solutionCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

To determine the consistency of the system of equations, we need to solve the system and check if there is a unique solution, infinitely many solutions, or no solution at all.

Let's write the given system of equations in matrix form:

```
| 1 2 -2 -1 | | x | | 0 |
| 2 -1 -1 0 | * | y | = | 0 |
| 1 2 0 -1 | | z | | 0 |
| 4 -1 -3 -1 | | u | | 0 |
```

We can represent this system as the augmented matrix `[A|B]`, where `A` is the coefficient matrix and `B` is the constant matrix.

To determine the consistency of the system, we need to perform row operations on the augmented matrix `[A|B]` and check the resulting matrix for any inconsistencies.

Let's perform row operations to bring the augmented matrix to its row-echelon form:

1. R2 = R2 - 2R1
```
| 1 2 -2 -1 | | x | | 0 |
| 0 -5 3 2 | * | y | = | 0 |
| 1 2 0 -1 | | z | | 0 |
| 4 -1 -3 -1 | | u | | 0 |
```

2. R3 = R3 - R1
```
| 1 2 -2 -1 | | x | | 0 |
| 0 -5 3 2 | * | y | = | 0 |
| 0 0 2 0 | | z | | 0 |
| 4 -1 -3 -1 | | u | | 0 |
```

3. R4 = R4 - 4R1
```
| 1 2 -2 -1 | | x | | 0 |
| 0 -5 3 2 | * | y | = | 0 |
| 0 0 2 0 | | z | | 0 |
| 0 -9 5 3 | | u | | 0 |
```

4. R4 = R4 + 9/5R2
```
| 1 2 -2 -1 | | x | | 0 |
| 0 -5 3 2 | * | y | = | 0 |
| 0 0 2 0 | | z | | 0 |
| 0 0 2 3 | | u | | 0 |
```

5. R4 = R4 - R3
```
| 1 2 -2 -1 | | x | | 0 |
|

Answer

Ni

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