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domain range A A -1 Pamela Leutwyler A Square matrix with 1’s on the diagonal and 0’s elsewhere Is called an IDENTITY MATRIX. I For every vector v, I.

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Presentation on theme: "domain range A A -1 Pamela Leutwyler A Square matrix with 1’s on the diagonal and 0’s elsewhere Is called an IDENTITY MATRIX. I For every vector v, I."— Presentation transcript:

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2 domain range A A -1 Pamela Leutwyler

3 A Square matrix with 1’s on the diagonal and 0’s elsewhere Is called an IDENTITY MATRIX. I For every vector v, I v = v

4 A square matrix A has an inverse if there is a matrix A -1 such that: AA -1 = I

5 Only one to one mappings can be inverted: v v R   v R Is the counterclockwise Rotation of through degrees. v v v RIf you know the value of You can find because Rotation is 1 – 1 (invertible) v P Is the projection of onto w v w v

6 Only one to one mappings can be inverted: v v R   v R Is the counterclockwise Rotation of through degrees. v v v RIf you know the value of You can find because Rotation is 1 – 1 (invertible) v P Is the projection of onto w v w v P is NOT 1-1. Given P v, v could be any one of many vectors vvv P is NOT invertible

7 Now we will develop an algorithm to find the inverse for a matrix that represents an invertible mapping.

8 A A -1 = I To solve for a, b, c, reduce: To solve for d, e, f, reduce: To solve for g, h, j, reduce:

9 To solve for a, b, c, reduce: To solve for d, e, f, reduce: To solve for g, h, j, reduce: It is more efficient to do the three problems below in one step

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11 It is more efficient to do the three problems below in one step 1 1 0

12 It is more efficient to do the three problems below in one step

13 It is more efficient to do the three problems below in one step

14 It is more efficient to do the three problems below in one step

15 A I I A -1 reduces to:


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