Determinant of 2 by 2 and 3 by 3 matrices (a Further Maths topic). Cramer's Rule (a First Year university topic).
Solving systems of equations using Cramer's Rule of Determinants.
Inverse of 2 by 2 and 3 by 3 matrices (a Further Maths topic).
Solving systems of equations using Gaussian elimination and LU factorisation; matrix inversion using Gauss-Jordan elimination. A First Year university topic.
Solving systems of equations by reducing a matrix to row echelon form using row operations.
Determined, overdetermined and underdetermined systems; unique solutions, and infinite and empty solution sets.
Inverting an \(n\) by \(n\) matrix by reducing to diagonal form using row operations.
Solving systems of equations by resolving a square matrix into lower-triangular and upper-triangular factors, followed by back- and forward-substitution.
Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors of square matrices. You may possible have met this if you've done Further Maths or the equivalent.
Eigenvalues and eigenvectors of a 2 by 2 matrix
Eigenvalues and eigenvectors of a 3 by 3 matrix.
Diagonalisation of a square matrix; the diagonal matrix of eigenvalues and the diagonalising matrix of eigenvectors.
The normalised eigenvectors of a symmetric matrix are orthonormal.
How to tell whether a matrix can be diagonalised.
A square matrix satisfies its own characteristic equation; using this fact to calculate powers and inverses.