The problem of solving n linear equations of n unknown can be reformulated as a task of finding a vector of n components such that when multiplied by given matrix it renders given right hand side vector. So for examle the system of equations 

                 3x + 4y   = 8
                 2x +  y   = 6
can be rewritten in matrix-vector form as 
                |3    4|   |x|   |8|
                |      | * | | = | |
                |2    1|   |y|   |6|
where 
     3  4
     2  1 is called matrix of the system and

     8
     6 is the right hand side vector.
Matrix of the system is called regular if there exists a matrix inverse to it, i.e. such that product of these matrices is a unity matrix. Unity matrix has all elements on main diagonal equal one and all others equal zero. From associativity of the multiplication of matrices and vectors then follows, that product of the inverse matrix and right hand side vector is the solution of the problem. If the matrix of the system is singular, i.e. does not have an inverse, the problem has either infinity many solutions or no solution at all. Applet running on the page you accessed this page from cannot solve a problem with a singular matrix, but it will recognize that the matrix of the system is singular and say so.