domain range A A-1 MATRIX INVERSE

I A Square matrix with 1’s on the diagonal and 0’s elsewhere Is called an IDENTITY MATRIX. 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: w v v  v P Is the projection of onto v v R Is the counterclockwise Rotation of through degrees. w v  v R If you know the value of You can find because Rotation is 1 – 1 (invertible)

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

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:

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:

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

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

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

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

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

A I reduces to: I A-1