Frequently Asked Questions
How are linear systems solved?
By transforming the equations using elementary operations (swap, scale, add) until the unknowns are isolated. Numerically this is Gaussian elimination; on paper it is substitution or elimination.
When is a system inconsistent?
When the coefficient matrix is singular and the right-hand side does not lie in the column space. After elimination one row reads 0 equals a nonzero number.
Why prefer Gaussian elimination?
It runs in n cubed time, generalizes to any size, and with partial pivoting is numerically stable. Cramer's rule is slow for big n; direct inverse is unstable.
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