# 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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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 v = v

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

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

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

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

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:

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

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

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

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

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

A I I A -1 reduces to:

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