## Contents

## Lecture 4 (01/21/10)

## Row Echelon Form (ref)

Definition:Let A be a matrix, A will be a row echelon form(ref) if:

1. If any, a row full of zeros has to be at the bottom.

2. The leftmost nonzero in any row is "1", also known as 'leading 1's'

3. If row i and j are nonzero and i<j, the the 'leading 1' in row i is to the left of'leading 1'in row j

Note:Numbers following the 'leading 1's' can be any numbers

## Reduced Row Echelon Form (rref)

Definition:Matrix A is in reduced row echelon form(rref) if:

1. A is in row echelon form(ref)

2. Any number above 'leading's 1' can only be zeros

Note:Every number in the column above the 'leading 1' need to be zeros

## Elementary Transformation Steps for rref Conversion

1. Switching rows

2. Scale rows with any number

3.Take any row and add a scale version of any other row to it.

Note:It is always a good idea to try to rearrange rows to make the matrix easier to convert first and tries to avoid creating any fractions, especially in the early stages.

Properties of a Determinant

1. det(A) = det(transpose of A)

2. det(A with rows i and j interchanged) = -det(A)

3. det(A with row j replaced by row j +c*rowi) = det(A)

4. det(matrix with a row of zeros) = 0

5. det(matrix with 2 equal rows) = 0

- All of these properties stay true if you replace row with column

6. det(AB) = det(A)*det(B)

7. if det(A) = 0 then it has no inverse

8. det(BA(inverseB)) = det(A)

Category:MA265Spring2010Walther