I have a problem with the question below:
2x_1 + x_2 + 3x_3 = 1
4x_1 + 6x_2 + 8x_3 = 5
6x_1 + ax_2 + 10x_3 = 5
where |a| < 10. For which of the following values of a will there be no row interchange required when solving this system using scaled partial pivoting?
a = 6, a = 9, a = -3
The answer in the book says a = 6, but I fail to see how any of the 3 values of a requires row interchange at all.
Here's my reasoning:
Largest coefficient in Row 1 = 3
Largest coefficient in Row 2 = 8
Largest coefficient in Row 3 = 10
2/3 = 0.67
4/8 = 0.5
6/10 = 0.6
In all 3 cases, a does not affect the results at all, which is the largest ratio is 0.67, so there is no need for row interchange.
What am I missing?