Applications of Matrices and Determinants
Applications of Matrices and Determinants tests whether you can reduce a matrix to echelon form, read its rank, and use rank or determinants to classify a linear system. The exam favours 3×3 systems and questions that ask for the number of solutions rather than only the solution itself.
Rank and consistency
The rank of a matrix is the number of nonzero rows in any row-echelon form. For AX=B with n unknowns, compare rank(A) with rank(A|B). Unequal ranks mean no solution. Equal ranks r mean consistent: r=n gives a unique solution, r<n gives infinitely many. For a square invertible A, X=A^{-1}B and Cramer's rule apply when det A≠0. A homogeneous system is always consistent; it has a nontrivial solution exactly when det A=0 for a square matrix, equivalently rank(A)<n.
Worked example
Solve x+y+z=3, 2x+2y+2z=6, x−y+z=1. The second row is twice the first, so rank(A)=2; the augmented matrix also has rank 2 because the same dependence holds. Since n=3, there are infinitely many solutions. From the first and third equations, x+z=3−y and x+z=1+y, so 3−y=1+y, y=1, and x+z=2. Let z=t; then x=2−t. The solution set is (2−t, 1, t).
Traps
Do not conclude inconsistency from det A=0; it only rules out a unique solution. Do not forget the augmented column when testing consistency. Row operations preserve rank but change determinant, so never compute a determinant after row operations and compare it with the original.