Matrix Algebra

An Introduction to Matrix Algebra 0011 0010 1010 1101 0001 0100 1011 Math 2240 Appalachian State University Dr. Ginn 4 1 2 Matrix Operations 0011 0010 1010 1101 0001 0100 1011 • 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. 4 1 2 Matrix Terminology 0011 0010 1010 1101 0001 0100 1011 • We say that two mn matrices A and B are equal if they have the same size and ai,j=bi,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. (ai) 4 1 2 Matrix Addition 0011 0010 1010 1101 0001 0100 1011 • If A and B are 2 m  n matrices then A+B is the m  n matrix with entries (a+b)i.j= ai,j+bi,j. 4 1 2 Scalar Multiplication 0011 0010 1010 1101 0001 0100 1011 • 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*ai,j. • With this definition we can define A-B to be A+(-1)B. 4 1 2 Matrix Multiplication 0011 0010 1010 1101 0001 0100 1011 • If A is an m  p matrix and B is an p  n matrix then AB is the m  n matrix with entries, p (ab)i, j  ai, k bk , j . k 1 Why???? 4 1 2 • Consider the system of equations, 0011 0010 1010 1101 0001 0100 1011 a1,1 x  a1,2 y  a1, 3 z  b1 a2,1 x  a2,2 y  a2,3 z  b2 a3,1 x  a3,2 y  a3,3 z  b3 • 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. 4 1 2 Partitioned Matrices 0011 0010 1010 1101 0001 0100 1011 • Sometimes it is also useful to think of a linear system in the following way: a11  a12  a1n  b1  a21  a22  a2n  b2  x1   x 2     x n                    am1  am 2  am n bm   4 1 2 • We say here that we have partitioned A into its column matrices a1, a2, ... , an, and 0011 0010 1010 1101 0001 0100 1011 written b as a linear combination of a1, a2, ... , an. • HMWK: p. 51: 5, 6, 8, 10, 14, 15, 16, 22, 24, 30, 32, 38 4 1 2