Engineering Mathematics · topic
Consistency of Ax = b by comparing rank A with rank [A | b], unique, infinite or no solution, homogeneous systems and free variables, Gaussian elimination and Cramer's rule; LU decomposition (Doolittle and Crout), when it exists, PA = LU and solving by forward and back substitution.
A system of m linear equations in n unknowns can have exactly one solution, infinitely many, or none. Rank decides which, and Gaussian elimination computes both the rank and the solutions. The rank-nullity theorem then counts how many free parameters a solution set has.
Gaussian elimination does two jobs at once: it transforms the matrix and it transforms the right-hand side. LU decomposition separates them.
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