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A. domain. range. A -1. MATRIX INVERSE. Pamela Leutwyler. For every vector v , I v = v. I. A Square matrix with 1’s on the diagonal and 0’s elsewhere Is called an IDENTITY MATRIX. A square matrix A has an inverse if there is a matrix A -1 such that: AA -1 = I. P. R.

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  1. A domain range A-1 MATRIX INVERSE Pamela Leutwyler

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

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

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

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

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

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

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

  9. -1 It is more efficient to do the three problems below in one step

  10. -1 It is more efficient to do the three problems below in one step 1 1 0 - 1

  11. -2 It is more efficient to do the three problems below in one step 0 1 -2 3

  12. -4 It is more efficient to do the three problems below in one step 0 7 -4

  13. -1 It is more efficient to do the three problems below in one step 3 -8 4 0

  14. A I reduces to: I A-1

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