Page 148 - Schaum's Outline of Differential Equations

CHAPTER 15

Matrices

MATRICES AND VECTORS

A matrix (designated h\ an uppercase boldface letter) is a rectangular arra\ of elements arranged in horizontal
rows and vertical columns. In this book, the elements of matrices will alwa\s be numbers or functions of the
variable ;. II all the elements are numbers. Ihen the matrix is called a constant matrix.
Matrices will prove to be very helpful in several ways. For example, we can recast higher-order differential
equations into a sjslem of first-order differential equations using matrices (see Chapter 17). Matrix notation
also provides a compact wa\ of expressing solutions to differential equations (see Chapter 16).

Example 15.1.

are all matrices. In particular, the first matrix is a constant matrix, whereas the last two are not.

A general matrix A haxing/j rows and n columns is gi\en h\

where a;; represents that element appearing in the /th row and /lh column. A matrix is square if it has the same
number of rows and columns.
A vector (designated bv a lowercase boldface letter) is a matrix having onl\ one column or one row. (The
third matrix given in Example I5.I is a vector.)

MATRIX ADDITION

The.viim A + B of two matrices A = |<;,,| and B = I/;,, having the same number of rows and the same number
of columns is the matrix obtained b\ adding the corresponding elements of A and B. That is,

Matrix addition is both associative and commutalue. Thus. A + (B + C) = (A + B) + C and A + B = B + A.

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