MAT 1234 Calculus I

MAT 1234Calculus I 9.5 The Algebra of Matrices http://myhome.spu.edu/lauw

HW • WebAssign 9.5

Preview • Look at the algebraic operations of matrices • “term-by-term” operations • Matrix Addition and Subtraction • Scalar Multiplication • Non-“term-by-term” operations • Matrix Multiplication

Matrix • If a matrix has m rows and n columns, then the size (dimension) of the matrix is said to be mxn.

Notations • Matrix

Special Cases • Row Vector • Column Vector

Matrix Addition and Subtraction • Let A = [aij] and B = [bij] be mxn matrices • Sum: A + B = [aij+bij] • Difference: A-B = [aij-bij] • (Term-by term operations)

Example 1

Scalar Multiplication Let A = [aij] be a mxn matrix and c a scalar. • Scalar Product: cA=[caij]

Example 2

Matrix Multiplication • Define multiplications between 2 matrices • Not “term-by-term” operations

Motivation • The LHS of the linear equation consists of two pieces of information: • coefficients: 2, -3, and 4 • variables: x, y, and z

Motivation • Since both the coefficients and variables can be represented by vectors with the same “length”, it make sense to consider the LHS as a “product” of the corresponding vectors.

Row-Column Product

Example 3

Matrix Multiplication

Example 4

Example 5 (a)

Example 5 (b)

Example 5 (c)

Example 5 (d)

Example 5 (e)

Example 5 (f)

Interesting Facts • The product of mxp and pxn matrices is a mxn matrix. • In general, AB and BA are not the same even if both products are defined. • AB=0 does not necessary imply A=0 or B=0. • Square matrix with 1 in the diagonal and 0 elsewhere behaves like multiplicative identity.

Identity Matrix nxn Square Matrix

Representation of Linear System by Matrix Multiplication

Remark It would be nice if “division” can be defined such that: (9.6) Inverse

MAT 1234 Calculus I