MATRIX ALGEBRA MGT 4850 Spring 2009 University of Lethbridge

Laws of Arithmetic Let A,B,C be matrices of the same size m × n, 0 the m × n zero matrix, and c and d scalars. (1) (Closure Law) A + B is an m × n matrix. (2) (Associative Law) (A + B) + C = A + (B + C) (3) (Commutative Law) A + B = B + A (4) (Identity Law) A + 0 = A (5) (Inverse Law) A + (−A) = 0 (6) (Closure Law) cA is an m × n matrix.

Laws of Arithmetic (II) (7) (Associative Law) c(dA) = (cd)A (8) (Distributive Law) (c + d)A = cA + dA (9) (Distributive Law) c(A + B) = cA + cB (10) (Monoidal Law) 1A = A

Matrix Multiplication Definition of Multiplication 2x − 3y + 4z = 5 as a “product” of the coefficient matrix [2,−3, 4] and the column matrix of unknowns x │ y │ z

Also example of vector multiplication!!!

Vector Multiplication

Vector Multiplication???

Matrix Multiplication Not Commutative or Cancellative

Identity matrix

Linear Systems as a Matrix Product

Ax=b

Laws of Matrix Multiplication Let A,B,C be matrices of the appropriate sizes so that the following multiplications make sense, I a suitably sized identity matrix, and c and d scalars. (1) (Closure Law) The product AB is a matrix. (2) (Associative Law) (AB)C = A(BC) (3) (Identity Law) AI = A and IB = B

Laws of Matrix Multiplication (4) (Associative Law for Scalars) c(AB) = (cA)B = A(cB) (5) (Distributive Law) (A + B)C = AC + BC (6) (Distributive Law) A(B + C) = AB + AC (skip from p.67 to p.101)

Matrix Inverses Let A be a square matrix. Then a (two- sided) inverse for Invertible A is a square matrix B of the same size as A such that AB = I = BA. If such Matrix a B exists, then the matrix A is said to be invertible. Application-if we could make sense of “1/A,” then we could write the solution to the linear system Ax = b as simply x = (1/A)b.

Singular = nonivertable Any nonsquare matrix is noninvertible. Square matrices are classified as either “singular,” i.e., noninvertible, or nonsingular,” i.e., invertible. Since we will mostly be concerned with two-sided inverses, the unqualified term “inverse” will be understood to mean a “two-sided inverse.” Notice that this definition is actually symmetric in A and B. In other words, if B is an inverse for A, then A is an inverse for B.

Examples of Inverses

Laws of Inverses (1) (Uniqueness) If A is invertible, then it has only one inverse, by A −1. (2) (Double Inverse) If A is invertible, then (A −1 ) −1 = A. (3) (2/3 Rule) If any two of the three matrices A, B, and AB are invertible, then so is the third, and moreover, (AB) −1 = B −1 A −1.

Laws of Inverses (4) If A is invertible, then (cA) −1 = (1/c)A −1. (5) (Inverse/Transpose) If A is invertible, then (A T ) −1 = (A −1 ) T. (6) (Cancellation) Suppose A is invertible. If AB = AC or BA = CA, then B = C. skip from p.103 to p.113

Basic Properties of Determinants

Cramer’s Rule Let A be an invertible n×n matrix and b an n×1 column vector. Denote by Bi the matrix obtained from A by replacing the ith column of A by b. Then the linear system Ax = b has unique solution x = (x1, x2,..., xn),

Example Use the Cramer’s rule to solve the system

Solution The coefficient matrix and right-hand- side vectors are