Inverse matrices and systems
Core study note
Current course guide + older Gauss–Jordan depth
Inverse by augmentation
For a square matrix A, augment with the identity: [A | I]. If row reduction reaches [I | B], then B = A⁻¹. If the left block cannot become the identity, A is singular.
System classification
A pivot in every variable column gives a unique solution. Missing variable pivots without contradiction produce infinitely many solutions. A contradiction row means the system is inconsistent.