8.3 Another Way of Solving a System of Equations Objectives: 1.) Learn to find the inverse matrix 2.) Use the inverse matrix to a system of equations

Consider this Let A=Y= B= Find Y if A + Y = B

Consider this Let A=Y= B= Find Y if AY = B

There is no division operation on matrices

Alternative Form for Solving a System of Equations Using the Inverse Matrix New Notation Let A be the cofficient matrix Let X be the variable matrix Let B be the solution matrix Thus, AX= B

Coefficient Matrix (A) A matrix whose real entries are the coefficients from a system of equations

Variable Matrix (X) A column matrix of the unknown variables

Solution Matrix A column matrix whose entries are the solutions of the system of equations

Identity Matrix A square matrix with a diagonal of 1s and all other entries are zeros RREF Form Notation: I

Characteristic of the Identity Matrix When a matrix is multiplied by the identity, you get the same matrix; AI= A

Example

Inverse Matrix Let A be a square matrix, then A -1 is the inverse matrix if AA -1 = I = A -1 A

Example A = B= Thus B can be notated A -1 because it is the inverse of A.

Finding the Inverse Matrix (The original matrix needs to be square!) 1.) Write the augmented matrix with [A:I] (The coefficient matrix and the identity matrix side by side 2.) Do proper row reductions to both A and I until A is in rref form (It has become an identity matrix itself 3.) The change in I is the inverse matrix of A, A -1 *** If you get a row of full zeros, the inverse does not exist****

Example Pg. 579 #22

Example: Find the inverse matrix of

How this helps us solve a system of equations. Example: Pg. 580 #53

Shortcut for finding the inverse of a 2x2 Pg. 577: If A is invertible if ad-bc ≠0 There is no inverse if ad-bc=0

A is invertible if ad-bc ≠0

Homework: 8.3 Page 579 # 2; 5; 19-22; 39-47(odd); 53; 54; 60; 71