Question

If \(AX = D\) represents the system of linear equations \[ 3x - 4y + 7z + 6=0,\quad 5x + 2y - 4z + 9=0,\quad 8x - 6y - z + 5=0, \] then AX = D ?

• A. \(\text{Rank}(A) = \text{Rank}([A|D])= 1\)
• B. \(\text{Rank}(A) = \text{Rank}([A|D])= 2\)
• C. \(\text{Rank}(A) = \text{Rank}([A|D])= 3\)
• D. \(\text{Rank}(A) \neq \text{Rank}([A|D])\)

Hint:

For an \(n\)-variable system, if \(\mathrm{Rank}(A) = \mathrm{Rank}([A|D])=n\), there is exactly one solution.

Answer 1

The Correct Option is C

Step 1: Form the augmented matrix and examine ranks.
These three linear equations in three unknowns typically give a \(\,3\times 3\) coefficient matrix \(A\) and a corresponding augmented matrix \([A\mid D]\). Step 2: Conclude about solution existence.
When \(\text{Rank}(A) = \text{Rank}([A|D])= 3\) in a 3-variable system, it implies the system has a unique solution (non-singular matrix). Hence \(\boxed{\text{Rank}(A) = \text{Rank}([A|D])= 3}\).