Discussion
Consider the simultaneous equations
\(2x + 4y = -7\); \(3x + 5y = 1\)
While solving using Gauss elimination without pivoting, after the forward elimination, the diagonal elements of the coefficient matrix are
Answer: A
The diagonal elements are 2 and \(-1\).
The coefficient matrix has rows (2, 4) and (3, 5). Without pivoting the first row stays as the pivot row, and the second row has \(3/2\) times the first subtracted from it to clear the \(x\) term:
\[ \begin{aligned} m &= \frac{3}{2} \\ a_{22} &= 5 - \frac{3}{2}(4) = 5 - 6 = -1 \end{aligned} \]
So the upper triangular matrix has rows (2, 4) and (0, \(-1\)), and its diagonal is 2 and \(-1\).
The words 'without pivoting' matter: partial pivoting would swap the rows to put the larger 3 on top, and the diagonal would then be 3 and \(4 - \frac{2}{3}(5) = \frac{2}{3}\), which is not offered either. The pairs 3 and 2, 5 and \(-2\), and 3 and 4 do not arise from eliminating this system; 3 and 4 simply mixes entries of the original matrix.
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