Matrices and Linear Algebra
Matrices and Linear Algebra tests matrix arithmetic, determinants, inverses and solving linear systems, with geometric interpretations of transformations.
Core rules
- Addition and multiplication require compatible orders; AB≠BA generally. (AB)^T=B^T A^T and (AB)^{-1}=B^{-1}A^{-1}.
- For 2×2 A=[[a,b],[c,d]], det A=ad-bc; A^{-1}=(1/det A)[[d,-b],[-c,a]] if det≠0.
- A system Ax=b has a unique solution x=A^{-1}b when det A≠0; det=0 means no unique solution.
- Matrix transformations act on position vectors; composition is matrix product in the order applied.
Worked example
Solve 2x+y=5, x-y=1. A=[[2,1],[1,-1]], det=-3. Then A^{-1}=(-1/3)[[-1,-1],[-1,2]]=[[1/3,1/3],[1/3,-2/3]]. Multiplying by [5,1]^T gives [2,1]^T, so x=2, y=1.
Traps
Multiplying in the wrong order; using ad-bc with wrong signs; inverting a det=0 matrix; forgetting row/column compatibility; applying transformation composition in reverse order; assuming AB=BA.