An Introduction to Matrix Algebra Math 2240 Appalachian State University Dr. Ginn

Matrix Operations Although we introduced matrices as a structure for convenient “bookkeeping” when solving systems of linear equations, they are interesting mathematically in their own right. We can define the operations of addition, scalar multiplication, subtraction and multiplication on them.

Matrix Terminology We say that two m  n matrices A and B are equal if they have the same size and a i,j =b i,j for 1≤ i ≤ m and 1 ≤ j ≤ n. A matrix with only one row is called a row matrix or row vector. A matrix with only one column is called a column matrix and column vector. (a i )

Matrix Addition If A and B are 2 m  n matrices then A+B is the m  n matrix with entries (a+b) i.j = a i,j +b i,j.

Scalar Multiplication If A is an m  n matrix and c is a scalar then cA is the m  n matrix with entries, (ca) i,j = c*a i,j. With this definition we can define A-B to be A+(-1)B.

Matrix Multiplication If A is an m  p matrix and B is an p  n matrix then AB is the m  n matrix with entries, Why????

Consider the system of equations, If A is the coefficient matrix of this system x is the column matrix of variables and b is the column matrix of right hand side numbers then the system is expressed as Ax = b.

Partitioned Matrices Sometimes it is also useful to think of a linear system in the following way:

We say here that we have partitioned A into its column matrices a 1, a 2,..., a n, and written b as a linear combination of a 1, a 2,..., a n. HMWK: p. 51: 5, 6, 8, 10, 14, 15, 16, 22, 24, 30, 32, 38