Question
Consider a system of linear equations
Ax = b,
where
A =
[
1 -√2 3
-1 √2 -3
]
b =
[
1
3
]
This system of equations admits
| Option | Description |
|---|---|
| (A) | a unique solution for x |
| (B) | infinitely many solutions for x |
| (C) | no solutions for x |
| (D) | exactly two solutions for x |
Solution
Step 1: Write the System
The given matrix equation is
A =
[
1 -√2 3
-1 √2 -3
]
b =
[
1
3
]
Therefore, the system of equations is
x₁ − √2 x₂ + 3x₃ = 1
−x₁ + √2 x₂ − 3x₃ = 3
Step 2: Compare the Two Rows of A
Observe that
(-1, √2, -3) = -(1, -√2, 3).
Thus, the second row of the coefficient matrix is exactly the negative of the first row. Therefore,
rank(A) = 1.
Step 3: Check Whether the System is Consistent
If the second row is the negative of the first row, then the second entry of the vector b should also be the negative of the first entry.
However,
b =
[
1
3
]
Since
3 ≠ -1,
the right-hand side is not consistent with the coefficient matrix.
Step 4: Verify by Multiplying the First Equation by −1
The first equation is
x₁ − √2x₂ + 3x₃ = 1.
Multiplying by −1 gives
−x₁ + √2x₂ − 3x₃ = −1.
But the second equation is
−x₁ + √2x₂ − 3x₃ = 3.
Since
−1 ≠ 3,
the two equations contradict each other. Therefore, the system is inconsistent.
(C) No solutions for x.