Matrices, determinants and systems of linear equations
Matrices, determinants and systems of linear equations
This shared MathematicsII study block practises matrix equations, determinants and parameter-dependent linear systems. It does not assert how often a particular regional PAU paper asks these questions.
Core rules
- Products and inverses: matrix product is not commutative; (AB)^T = B^T A^T and (AB)^(-1) = B^(-1)A^(-1). From AX = B with |A| != 0, X = A^(-1)B; from XA = B, X = BA^(-1). The side on which you multiply matters.
- Determinants: |AB| = |A||B|, |A^T| = |A|, |A^(-1)| = 1/|A|, and |kA| = k^n|A| for an n x n matrix. A zero row, two equal rows, or a row that is a linear combination of the others makes the determinant zero.
- Inverse by adjoint: A^(-1) = adj(A)/|A|, where adj(A) denotes the transpose of the cofactor matrix, valid only when |A| != 0.
- Rank: the number of non-zero rows left after Gaussian elimination; elementary row operations preserve it.
- Rouche-Capelli: A x = b is consistent iff rank(A) = rank(A|b). It has a unique solution iff that common rank equals n, the number of unknowns; otherwise there are n - rank free parameters.
- Homogeneous systems are always consistent, and have non-trivial solutions exactly when rank(A) < n.
Worked example
Discuss for which values of k the system x + y + z = 1, x + 2y + 3z = 2, x + 4y + kz = 4 has solutions.
The coefficient determinant is |A| = 1(2k - 12) - 1(k - 3) + 1(4 - 2) = k - 7. If k is not 7 the system is determined and Cramer gives a unique solution. For k = 7, subtracting row 1 from rows 2 and 3 leaves (0,1,2|1) and (0,3,6|3), and the third row is three times the second, so rank(A) = rank(A|b) = 2 < 3. The system is consistent with one parameter: setting z = L gives y = 1 - 2L and x = L.
Traps that cost marks
Using |kA| = k|A| instead of k^n|A|; applying Cramer when the determinant vanishes; concluding that rank(A) = rank(A|b) implies a unique solution (you also need rank = n); forgetting that rank(A) < n gives infinitely many solutions only when rank(A) = rank(A|b); otherwise the system is inconsistent; and multiplying by the inverse on the wrong side in a matrix equation.