1. domain
range A
A-1
MATRIX INVERSE

2. 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

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

4. 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)

5. 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

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

7. 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:

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

10. It is more efficient to do the three problems below in one step1 1
- 1 -1

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

12. It is more efficient to do the three problems below in one step7 -4 -4

13. It is more efficient to do the three problems below in one step3 -8 4 -1

14. A I
reduces to:
I A-1