4.4 & 4.5 Notes

4.4 & 4.5 Notes

Identity MatricesRemember:

IdentityMatrices:

If the product of two matrices equal the identity matrix then they are inverses.IDENTITY MATRIX PROOF

a = (-3)(1) + (4)(0) = -3b = (-3)(0) + (4)(1) = 4c = (-2)(1) + (6)(0) = -2d = (-2)(0) + (6)(1) = 6

The Inverse formula of a 2x2 Matrix

Step 2) Switch a & dStep 3) Change the signs of b&cStep 1) Find determinant A scalar, Put under 1

Step 4) Multiply scalarFIND THE INVERSE

Step 1: Find Determinant A (scalar), put under 1

Step 2:Step 3:SWITCH 5 AND -2Change signs of 3 and 1

Step 4: Multiply scalar

Answer:1. Find the inverse

Step 1Answer =Steps 2 & 3Steps 4: Multiply scalarSolving Matrix EquationsFind the inverse of the matrix next to the variable

Multiply both sides by the inverse matrix, the inverse must be on the left side when multiplying-Check for the Identity matrix

Step 1:Find the Inverse Matrix First

(Multiply both sides by the inverse matrix on the left)Step 2

Multiply rows by columnsMultiply rows by columns

SolutionMultiply both sides by the inverse matrix)

Find the inverse first!!!!

Solve the Matrix Equation

Subtract Matrix from both sidesFind the inverseMultiply inverse by both sides (keep it left)

Homework:Read section 4.4***Define Identity and Inverse MatricesPgs. 227-229; 1-3, 14-32e, 54-60e

4.5 Solving systems using matrices.

A system can be written as a single matrix equation.

Linear systemMatrix equationMatrix A is called the Coefficient matrix.Matrix X is called the Variable matrixMatrix B is called the Constant matrixA X = BSolving for x and y

Step 2: Find the inverse of the Coefficient Matrix and multiply both sides

Step 1: Set up the equation in matrix form

Step 2: Finding the Inverse Matrix

Step 2: Multiply both sides by the Inverse(2,-2)

USE AN INVERSE MATRIX TO SOLVE THE LINEAR SYSTEM.

FIRST, BEGIN BY WRITING THE EQUATIONS IN MATRIX FORM.

SECOND, YOU MUST NOW FIND THE INVERSE OF THE COEFFICIENT MATRIX.FIND THE INVERSE OF THE COEFFICIENT MATRIX.

SOLVE THE SYSTEM BY MULTIPLYING BY THE INVERSE

x = 1 AND y = 2OR(1,2)

More Practice1.

3.2.

4.(8,4)(4,4)(-1,-5)(44/5, -26/5)HomeworkRead section 4.5***Matrix of variables and Matrix of constantsPg.233-235; 1-3, 12-18e, 24-30e, 48-62e