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.
The rank test for consistency of Ax = b (unique, infinite or no solution) for square and rectangular systems, homogeneous systems and free variables, Gaussian elimination on the augmented matrix with a worked infinite-solution example, a parameter system classified for every value, Cramer's rule and its limits, the structure of the solution set, geometric pictures, and two solved past questions with a parameter in a 2 x 2 system.
A partial preview of the top of the sheet. The full sheet is in the Engineering Mathematics cheat sheet pack.
What LU decomposition is and how it stores Gaussian elimination, Doolittle (unit lower) and Crout (unit upper) forms with entry formulas, a full 3 x 3 Doolittle and Crout factorisation of the same matrix, solving Ax = b by forward and back substitution, existence and uniqueness through leading principal minors, a matrix with no LU and the PA = LU fix, a singular matrix that still has an LU, operation counts, and Cholesky in one line. Every factor is checked by exact arithmetic.
A partial preview of the top of the sheet. The full sheet is in the Engineering Mathematics cheat sheet pack.
