# 9/16/2008

## Inverse of a Linear Transformation

- Invertibility and rref(A)
- Finding the inverse of a matrix

## Proof of the general form of the inverse for a 2x2 matrix

- Determinant of a 2x2 matrix

## Matrix Multiplication

- To find BA, we can multiply B with the columns of A and combine the resulting vectors, as long as the dimensions of A and B permit multiplication.
- Matrix multiplication is noncommutative
- Multiplying with the inverse
- Multiplying with the identity matrix
- Matrix multiplication is associative
- The inverse of a product of matrices
- A criterion for invertibility
- Distributive property for matrices
- A(C+D) = AC + AD
- (A+B)C = AC + BC

- If k is a scalar, then (kA)B = A(kB) = k(AB)
- Partitioned matrices

## Please "review" Appendix A

- Vector additon
- Scalar multiplication
- Rules for vector algebra
- Dot product
- v.w = v1 x w1 + v1 x w2 + ... + vn x wn

- Rules for dot products
- Length (or norm) of a vector
- Unit vector
- Perpendicular (or orthogonal) vectors
- Two vectors, v and w, are called perpendicular if their dot product is equal to zero.

- Cross product

## Please read Section 3.1 for class on Thursday