4.4 & 4.5 notes
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4.4 & 4.5 Notes. Identity Matrices. Remember :. Identity Matrices :. If the product of two matrices equal the identity matrix then they are inverses. IDENTITY MATRIX PROOF. a = (-3)(1) + (4)(0) = -3 b = (-3)(0) + (4)(1) = 4 c = (-2)(1) + (6)(0) = -2 d = (-2)(0) + (6)(1) = 6. - PowerPoint PPT PresentationTRANSCRIPT
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