MATRICES

Cayley, a British mathematician discovered matrices in the year 1860. But it was not until the twentieth century was well advanced that engineers heard of them. A matrix is a rectangular array of numbers. These days, however, such arrays (matrices) have been found to be of great utility in many branches of applied mathematics such as algebraic and differential equations, mechanics, theory of electric circuits, nuclear physics, aerodynamics and astronomy. In many cases, they form the coefficients of linear transformations or systems of linear equations arising, for instance, from electric network, frameworks in mechanics, curve fitting in statistics and transportation problems.

Matrices of the same size can be added or subtracted element by element. But the rule for matrix multiplication is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. A major application of matrices is to represent linear transformation, that is, generalizations of linear functions such as f(x) = 4x. For example, the rotation of vectors in three dimensional space is a linear transformation. If R is a rotation matrix and v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of a system of linear equations. If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero. Eigenvalues and eigenvectors provide insight into the geometry of linear transformations.

Matrices are useful because they enable us to consider an array of many numbers as a single object, denote it by a single symbol, and perform calculations with these symbols in a very compact form. The mathematical shorthand thus obtained is very elegant and powerful and is suitable for various practical engineering problems. It entered engineering mathematics over seventy years ago and is of increasing importance of various engineering branches. Therefore, it is necessary for the young engineers to learn the elements of matrix algebra in order to keep up with the fast development of physics and engineering.