Applications of Matrices and Determinants
The Applications of Matrices and Determinants domain tests whether you can reduce a matrix to row-echelon form, read its rank, and use that rank to classify a linear system as uniquely solvable, having infinitely many solutions, or inconsistent. It also tests the adjoint method for inverses and Cramer's rule.
Rank and consistency
Rank is the number of non-zero rows in an echelon form, or the order of the largest submatrix with non-zero determinant. For AX = B with n unknowns: rank(A) = rank([A|B]) = n gives a unique solution; rank(A) = rank([A|B]) < n gives infinitely many solutions with n − rank free variables; rank(A) ≠ rank([A|B]) means inconsistent. For AX = 0 a non-trivial solution exists exactly when det A = 0.
Inverses
A⁻¹ = adj A / det A, valid only when det A ≠ 0, and (AB)⁻¹ = B⁻¹A⁻¹. The adjoint is the transpose of the cofactor matrix, so transpose before dividing.
Worked example
For x + y + z = 1, x + 2y + 3z = 4, x + 3y + kz = 7 the coefficient determinant expands to (2k − 9) − (k − 3) + 1 = k − 5. If k ≠ 5 it is non-zero, rank = 3 and the solution is unique. If k = 5 the third equation equals twice the second minus the first, so rank(A) = rank([A|B]) = 2 < 3: consistent with 3 − 2 = 1 free variable, hence infinitely many solutions.
Traps
Finding rank(A) but not the augmented rank; declaring 'no solution' whenever the determinant vanishes; sign errors in cofactors; forgetting the zero solution always satisfies a homogeneous system.