Question

Consider the simultaneous equations
\[ 2x + 4y = -7; \quad 3x + 5y = 1 \] While solving using Gauss elimination without pivoting, after the forward elimination, the diagonal elements of the coefficient matrix are

• A. 2 and -1
• B. 3 and 2
• C. 5 and -2
• D. 3 and 4

Hint:

Eliminate x from the second equation using the first row's pivot, then look at the resulting coefficient of y.

Answer 1

The Correct Option is A

Step 1: Write the coefficient matrix.
The system is \( 2x + 4y = -7 \) and \( 3x + 5y = 1 \), so the coefficient matrix is
\[ \begin{bmatrix} 2 & 4 \\ 3 & 5 \end{bmatrix} \]
Step 2: Find the elimination factor.
Without pivoting, the first row stays as is, and the first entry \( 2 \) is the pivot.
The factor to clear the \( 3 \) in row 2 is \( m = \dfrac{3}{2} \).

Step 3: Apply the row operation.
New row 2 equals row 2 minus \( m \) times row 1:
\( 3 - \dfrac{3}{2}(2) = 0 \) for the first entry, as expected.
\( 5 - \dfrac{3}{2}(4) = 5 - 6 = -1 \) for the second entry.

Step 4: Read off the diagonal.
After forward elimination, the coefficient matrix becomes
\[ \begin{bmatrix} 2 & 4 \\ 0 & -1 \end{bmatrix} \] so the diagonal elements are \( 2 \) and \( -1 \).

Final Answer:
The diagonal elements after forward elimination are 2 and -1. \[ \boxed{2, \ -1} \]