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# Matrices (Metric)

Learn, or revise, determinants, matrix inverses, Gaussian elimination and eigenvalues/eigenvectors.

## Units

### Determinants

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.

### Inverses

Inverse of 2 by 2 and 3 by 3 matrices (a Further Maths topic).

### Gaussian Elimination

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.