Algebra
Algebra
This domain tests matrix algebra and determinants: conformability, multiplication, transpose, inverse, and solving small linear systems. ASSEB problems often require a 2×2 or 3×3 determinant and then an inverse or consistency check.
Core rules
- Matrices add only when same order. Product AB is defined when columns of A equal rows of B; entry (i,j) is row i of A times column j of B.
- Matrix multiplication is not commutative: AB≠BA in general. Transpose reverses order: (AB)^T=B^T A^T.
- For A=[[a,b],[c,d]], |A|=ad−bc. A is singular if |A|=0; then A^{-1} does not exist.
- For a 2×2 matrix, adj A=[[d,−b],[−c,a]] and A^{-1}=(1/|A|) adj A.
- (AB)^{-1}=B^{-1}A^{-1}, and |AB|=|A||B|.
Worked example
Let A=[[2,3],[1,4]]. Then |A|=2·4−3·1=8−3=5≠0, so A is invertible. adj A=[[4,−3],[−1,2]]. Hence A^{-1}=(1/5)[[4,−3],[−1,2]]. Verify: A A^{-1}=(1/5)[[2,3],[1,4]][[4,−3],[−1,2]]=(1/5)[[8−3,−6+6],[4−4,−3+8]]=(1/5)[[5,0],[0,5]]=I. This check is fast and catches sign errors in the adjoint.
Traps
Do not multiply corresponding entries; use row-by-column. In the adjoint, cofactors carry signs, and the adjoint is the transpose of the cofactor matrix. A zero determinant means no inverse, not a zero matrix. When solving AX=B, use X=A^{-1}B only if A is invertible; otherwise test consistency.