1.3 Reduced Row-Echelon Form

Define row-equivalent matrices

one can be obtained from the other by a sequence of row operations

Define augmented matrix

suppose we have a system of m equations in n variables, with coefficient matrix A and vector of constants b; the m × (n + 1) matrix whose first n columns are the columns of A and whose last column (n + 1) is the column vector b

Define zero vector ( of size m)

the column vector of size m where each entry is the number zero

Define leading 1

the leftmost nonzero entry of a nonzero row

Theorem RREFU

(Reduced Row-Echelon Form is Unique); Suppose that A is an m × n matrix and that B and C are m × n matrices that are row-equivalent to A and in reduced row-echelon form. Then B = C.

Theorem REMES

(Row-Equivalent Matrices represent Equivalent Systems); Suppose that A and B are row-equivalent augmented matrices. Then the systems of linear equations that they represent are equivalent systems.

Theorem REMEF

(Row-Equivalent Matrix in Echelon Form); Suppose A is a matrix. Then there is a matrix B so that 1. A and B are row-equivalent 2. B is in reduced row-echelon form

2. Use row operations to convert the matrix below to reduced row-echelon form and report the final matrix.

**on loose leaf paper

3. Find all the solutions to the system below by using an augmented matrix and row operations. Report your final matrix in reduced row-echelon form and the set of solutions.

**on loose leaf paper

A matrix is in reduced row-echelon form if it meets all of the following conditions:

1. If there is a row where every entry is zero, then this row lies below any other row that contains a nonzero entry. 2. The leftmost nonzero entry of a row is equal to 1. 3. The leftmost nonzero entry of a row is the only nonzero entry in its column. 4. Consider any two different leftmost nonzero entries, one located in row i, column j and the other located in row s, column t. If s > i, then t > j.

1. Is the matrix below in reduced row-echelon form? Why or why not?

It is because in each row, the first nonzero number is a 1. The ones go diagonally downward starting from the first row. Each 1 has a 0 above it.

Define pivot column

a column containing a leading 1

Define m × n matrix

a rectangular layout of numbers from the complex numbers having m rows and n columns

Define zero row

a row of only zero entries

Define column vector (of size m)

an ordered list of m numbers, which is written in order vertically, starting at the top and proceeding to the bottom

Define row operation

will transform an m × n matrix into a different matrix of the same size; 1. Swap the location of two rows. 2. Multiply each entry of a single row by a nonzero quantity. 3. Multiply each entry of one row by some quantity, and add these values to the entries in the same columns of a second row. Leave the first row the same after this operation, but replace the second row by the new values.