## Arrays Matrices and Determinants

Spreadsheet calculations lend themselves almost automatically to the use of arrays of values. Arrays in Excel can be either one- or two-dimensional. For the solution of many types of problem, it is convenient to manipulate an entire rectangular array of values as a unit. Such an array is termed a matrix. (In Excel, the terms "range," "array" and "matrix" are virtually interchangeable.) Anmxn matrix (m rows and n columns) of values is illustrated below:

The values comprising the array are called matrix elements. Mathematical operations on matrices have their own special rules, to be discussed in the following sections.

Some Types of Matrices

A matrix which contains a single column of m rows or a single row of n columns is called a vector.

A square matrix has the same number of rows and columns. The set of elements aij for which i = j {an, a22,..., a„„) is called the main diagonal or principal diagonal.

If all the elements of a square matrix are zero except those on the main diagonal, the matrix is termed a diagonal matrix. A diagonal matrix whose diagonal elements are all 1 is a unit matrix.

An upper triangular matrix has values on the main diagonal and above, but the values of all elements below the main diagonal are zero; similarly, a lower triangular matrix has zero values for all elements above the main diagonal.

A tridiagonal matrix contains all zeros except on the main diagonal and the two adjacent diagonals.

A symmetric matrix is a square matrix in which a^ = ap.

A determinant is a property of a square matrix; there is a procedure for the numerical evaluation of a determinant, so that an N x N matrix can be reduced to a single numerical value. The value of the determinant has properties that make it useful in certain tests and equations. (See, for example, "Cramer's Rule" in Chapter 9.)