## 15.1 Synopsis

Sets of linear equations in several variables can be described by their coefficients, which can be formed into a regular array, called a matrix; the operations used to solve equations can then be phrased in terms of matrices.
Matrices also have use in describing linear transformations defined on vectors: these transformations are operations which take vectors into vectors and are linear in that they take a weighted sum of vectors into the same weighted sum of what they take the vectors themselves into. Such transformations play an important role in physics.
The natural operations of addition and composition among transformations (you compose two transformations by performing one and then the other; you add them by forming the transformation whose output is the sum of theirs) give rise to natural definitions for the addition and multiplication of matrices.
We will discuss use of matrices to solve or reduce equations; define the determinant of a square matrix and describe its evaluation; describe the operations of addition and multiplication of matrices, and the concepts of inverse, singular matrix, eigenvalue, and eigenvector, and discuss how these can be found.