Tutorials: Matrices

by mathematics.me.uk

The following tutorials are free to download, print and distribute, so long as the web source is not deliberately deleted, or is otherwise acknowledged. Please contact if you have any questions.

Basics

A matrix (plural matrices) is an rectangular array of numbers enclosed by brackets. A vector is an array that consists of a single column or a single row. The following documents cover the definitions and basic operations on matrices.

• Matrix Definitions : Exercises
• Matrix Arithmetic : Exercises

Identity and Inverse Matrices

The identity matrix has the property that it doesn't change any matrix following multiplication. If two matrices multiplied together give the identity matrix, the matrices are mutually inverse. The following documents cover the definitions of the identity and inverse matrices. Methods for finding the inverse of 2x2 and 3x3 matrices is considered. Excel spreadsheets for finding the inverse of matrices can also be downloaded.

• Identity and Inverse Matrices :
• Inverse of a 2x2 Matrix : Spreadsheet: 2x2 Matrix Inverse Matrices
• Inverse of a 3x3 Matrix : Spreadsheet- Inverting a 3x3 Matrix
• Inverse of a 3x3 Matrix by elimination : Excel Spreadsheet on inverting a 3x3 Matrix by elimination

Solution of a Linear System of Equations

A system of linear algebraic equations has the form

a[1,1] x[1] + a[1,2] x[2] + .... + a[1,N] x[N] = b[1]
a[2,1] x[1] + a[2,2] x[2] + .... + a[2,N] x[N] = b[2]
a[M,1] x[1] + a[M,2] x[2] + .... + a[M,N] x[N] = b[M]

where the a[i,j] and the b[i] are known. The x[j] are unknown and the purpose of solving the system is to find the x[j].

The system is often written in the form A x = b where A is an MxN matrix and x is an N-vector and b is an M-vector.

The problem of solving A x = b - that is finding the value of the vector x for a given matrix A and a vector b is a fundamental problem in mathematics. In the following documents methods for solving such problems with 2x2 matrices and 3x3 matrices are considered.

• Linear Systems and 2x2 Matrices
• Spreadsheet: Solution of a 2x2 System by inverting the matrix
• Spreadsheet - Solving a 3x3 matrix-vector system

Eigenvalues and Eigenvectors

In solving matrix-vector problems, we generally find that the solution takes on characteristic forms. Hence the properties of the matrices - the eigenvalues and eigenvectors - are studied.

• Matrix Eigenvalues and Eigenvectors
• Generalised Eigenvalue Problem

Other Stuff

A vector norm is a measure of the size of a vector.

• Vector Norm

Matlab/ Freemat / Scilab / Octave

Matlab, Freemat, Scilab and Octave are computer programs that use arrays as a basic type and the operations carried out on matrices covered here can be carried out automatically and easily in the Matlab/Freemat environment. Freemat is free to download. Further information on Matlab and Freemat, including tutorials can be found on the freemat.info website.

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