Matrices, Determinants and Systems of Linear Equations
This domain covers matrix algebra (sum, product, transpose, powers, inverse), determinants up to order 3, rank, and the discussion and solution of linear systems. In the Galician PAU for Matemáticas Aplicadas ás Ciencias Sociais II the recurring items are a matrix equation such as AX + B = C, a system depending on a parameter a that must be discussed with the Rouché–Capelli theorem, and a 2×2 or 3×3 system solved by Cramer's rule or Gaussian elimination.
Matrix operations
- The product AB exists only when the number of columns of A equals the number of rows of B; the dimensions combine as (m×n)(n×p) = m×p.
- Generally AB ≠ BA, so never move a matrix to the other side of a product carelessly.
- (AB)^T = B^T A^T and, when both are invertible, (AB)⁻¹ = B⁻¹A⁻¹.
- A is invertible exactly when det A ≠ 0; then A⁻¹ = adj(A)/det A, where adj(A) is the transpose of the cofactor matrix.
- From AX = B multiply on the left: X = A⁻¹B. From XA = B multiply on the right: X = BA⁻¹.
Determinants and rank
det(AB) = det A · det B, det(A^T) = det A, det(kA) = kⁿ det A for an n×n matrix, and swapping two rows changes the sign. Rank is the order of the largest non-zero minor, and also the number of non-zero rows left after Gaussian elimination.
Discussing systems (Rouché–Capelli)
With A the coefficient matrix and A* the augmented matrix: rank A ≠ rank A* gives no solution; rank A = rank A* = n gives a unique solution; rank A = rank A* = r < n gives infinitely many solutions depending on n − r parameters.
Worked example
Solve 2x + y = 4, 5x + 3y = 11. Here det A = 6 − 5 = 1 ≠ 0, so Cramer applies: x = det[[4,1],[11,3]]/1 = 12 − 11 = 1, and y = det[[2,4],[5,11]]/1 = 22 − 20 = 2. Checking in the second equation: 5·1 + 3·2 = 11, correct.
Traps that cost marks
- Applying Cramer to a parameter system without isolating the values of a for which det A = 0; those cases need their own Rouché–Capelli discussion.
- Writing X = BA⁻¹ when the equation is AX = B, or forgetting to verify that A⁻¹ exists.
- Sign errors in cofactors: each cofactor carries the factor (−1)^(i+j).
- Answering "infinitely many solutions" without the parametric form when the question asks for the solutions themselves.