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Published byArthur Russell Modified over 6 years ago

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4.4 & 4.5 Notes

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Remember: Identity Matrices: If the product of two matrices equal the identity matrix then they are inverses.

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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

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Step 2) Switch a & d Step 3) Change the signs of b&c Step 1) Find determinant A scalar, Put under 1 Step 4) Multiply scalar

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

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Step 2: Step 3: SWITCH 5 AND -2 Change signs of 3 and 1 Step 4: Multiply scalar Answer:

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1. Find the inverse Step 1 Answer = Steps 2 & 3 Steps 4: Multiply scalar

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Solving Matrix Equations 1.Find the inverse of the matrix next to the variable 2.Multiply both sides by the inverse matrix, the inverse must be on the left side when multiplying -Check for the Identity matrix

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Step 1: Find the Inverse Matrix First

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

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Multiply both sides by the inverse matrix) Find the inverse first!!!! Solve the Matrix Equation

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1.Subtract Matrix from both sides 2.Find the inverse 3.Multiply inverse by both sides (keep it left)

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

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4.5 Solving systems using matrices.

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A system can be written as a single matrix equation. Linear system Matrix equation Matrix A is called the Coefficient matrix. Matrix X is called the Variable matrix Matrix B is called the Constant matrix A X = B

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Solving 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

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Step 2: Finding the Inverse Matrix

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

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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.

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FIND THE INVERSE OF THE COEFFICIENT MATRIX.

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SOLVE THE SYSTEM BY MULTIPLYING BY THE INVERSE x = 1 AND y = 2 OR (1,2)

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More Practice 1. 3. 2. 4. (8,4)(4,4) (-1,-5) (44/5, -26/5)

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Homework Read section 4.5 ***Matrix of variables and Matrix of constants Pg.233-235; 1-3, 12-18e, 24-30e, 48-62e

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