Matrices and Determinants
Matrices and Determinants
UPMSP tests conformability, matrix algebra, determinant properties, and solving small linear systems. A matrix of order m×n has m rows and n columns; addition requires the same order, while AB requires columns of A to equal rows of B. Matrix multiplication is row-by-column and is not commutative.
Core rules
- Transpose swaps rows and columns; (AB)^T=B^T A^T.
- Determinants exist only for square matrices. For A=[[a,b],[c,d]], |A|=ad-bc.
- |AB|=|A||B|; a determinant changes sign when two rows or columns are interchanged.
- A^{-1}=adj(A)/|A|, provided |A|≠0. If |A|=0, A is singular and has no inverse.
- For AX=B with |A|≠0, X=A^{-1}B; Cramer’s rule is an equivalent determinant method.
Worked example
Let A=[[2,1],[3,4]]. Then |A|=2(4)-1(3)=5, so A is invertible. For a 2×2 matrix, adj(A)=[[4,-1],[-3,2]], hence A^{-1}=(1/5)[[4,-1],[-3,2]]. To solve 2x+y=5, 3x+4y=11, write X=A^{-1}[5,11]^T=(1/5)[20-11, -15+22]^T=(9/5,7/5). Check: 2(9/5)+7/5=5 and 3(9/5)+4(7/5)=11.
Traps
Multiplying in the wrong order; replacing adj(A) with the cofactor matrix without transposing; dropping a minus sign in a 2×2 determinant; using Cramer’s rule when the determinant is zero; confusing infinite solutions with no solution.