Linear algebra is a very important area of mathematics, with numerous scientific applications. The most basic concept in linear algebra is the concept of a matrix. A matrix is a rectangular array of numbers. There is an important quantity associated with square matrices, known as the determinant, which if nonzero, implies that the matrix has a well-defined inverse.

Linear algebra is primarily concerned with solving systems of linear equations. Such a system may be represented as a matrix equation of the form Ax = b, where A is the matrix of coefficients of the equations, x is a column vector containing all the unknown quantities of the equations, and b is the column vector of constant terms of the equations. There are several methods for solving such a system of equations, each involving matrix operations. If A is an invertible square matrix, then the system has a unique solution of the form x = A^-1 B, where A^-1 is the inverse of A.

Although the above equation can always be used to solve a system of n linear equations in n variables, it is usually impractical to do so directly. There are much more efficient methods, which do not require computing the inverse explicitly. The fastest method is Gaussian elimination, which is a form of row reduction. The idea is to perform a sequence of linear operations on the rows of the augmented matrix [A|b], formed by adding the column vector b to the right side of the matrix A. When the process is complete, we are left with the matrix [I|x], where I is the identity matrix and x is the column vector of solutions.

Although matrices are primarily used for solving systems of linear equations, they have many other uses as well. Another application of matrices is in performing linear transformations of coordinates. These include reflections, rotations, stretches and shears.