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.
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.
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.
A system of linear algebraic equations has the form
a[1,1] x + a[1,2] x + .... + a[1,N] x[N] = b
a[2,1] x + a[2,2] x + .... + a[2,N] x[N] = b
a[M,1] x + a[M,2] x + .... + 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.
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.
Return to Mathematics Tutorials index .