Much of finite elements revolves around forming matrices and solving systems of linear equations using matrices. This learning resource gives you a brief review of matrices.
Contents 
Suppose that you have a linear system of equations
Matrices provide a simple way of expressing these equations. Thus, we can instead write
An even more compact notation is
Here is a matrix while and are matrices. In general, an matrix is a set of numbers arranged in m rows and n columns.
Common types of matrices that we encounter in finite elements are:
diagonal elements (a_{ii}) nonzero.
with each of its nonzero elements (a_{ii}) equal to 1.
such that a_{ij} = a_{ji}.
such that a_{ij} = − a_{ji}.
Note that the diagonal elements of a skewsymmetric matrix have to be zero: .
Let and be two matrices with components a_{ij} and b_{ij}, respectively. Then
Let be a matrix with components a_{ij} and let λ be a scalar quantity. Then,
Let be a matrix with components a_{ij}. Let be a matrix with components b_{ij}.
The product is defined only if n = p. The matrix is a matrix with components c_{ij}. Thus,
Similarly, the product is defined only if q = m. The matrix is a matrix with components d_{ij}. We have
Clearly, in general, i.e., the matrix product is not commutative.
However, matrix multiplication is distributive. That means
The product is also associative. That means
Let be a matrix with components a_{ij}. Then the transpose of the matrix is defined as the matrix with components b_{ij} = a_{ji}. That is,
A important identity involving the transpose of matrices is
The determinant of a matrix is defined only for square matrices.
For a matrix , we have
For a matrix, the determinant is calculated by expanding into minors as
In short, the determinant of a matrix has the value
where M_{ij} is the determinant of the submatrix of formed by eliminating row i and column j from .
Some useful identities involving the determinant are given below.
Let be a matrix. The inverse of is denoted by and is defined such that
where is the identity matrix.
The inverse exists only if . A singular matrix does not have an inverse.
An important identity involving the inverse is
since this leads to:
Some other identities involving the inverse of a matrix are given below.
determinant of its inverse.
is equal to the original matrix.
We usually use numerical methods such as Gaussian elimination to compute the inverse of a matrix.
A thorough explanation of this material can be found at Eigenvalue, eigenvector and eigenspace. However, for further study, let us consider the following examples:
Which vector is an eigenvector for ?
We have , and
Thus, is an eigenvector.
We have that since , is not an eigenvector for
