Row Reduced Echelon Form

Week 5: section 2.3

In Row Reduced Echelon Form, if there is a row of all zeros, where should it be?

At the bottom of the matrix

What is a leading 1?

The first non-zero entry in a row of a matrix that’s always 1

What does “pivots” refer to in Row Reduced Echelon Form?

The step patterns in which the leading ones must be positioned in their column

In Row Reduced Echelon Form, the entries above and below each leading 1 should be equal to what?

zero

True or false that in Row Reduced Echelon Form, the entries to the LEFT of each leading 1 should be equal to what?

zero (leading 1 must be the first on-zero entry)

True or false that in Row Reduced Echelon Form, the entries to the RIGHT of each leading 1 should be equal to what?

any number as long as it does not interfere with the column rule

Is the following matrix in Row Reduced Echelon Form? If not, explain why

[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]

yes

Is the following matrix in Row Reduced Echelon Form? If not, explain why

[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 0 ]

yes

Is the following matrix in Row Reduced Echelon Form? If not, explain why

[ 1 2 0 ]
[ 0 0 0 ]

yes

Is the following matrix in Row Reduced Echelon Form? If not, explain why

[ 1 0 0 0 ]
[ 0 0 0 1 ]

yes

Is the following matrix in Row Reduced Echelon Form? If not, explain why

[ 0 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]

no because the zero row must be at the bottom

Is the following matrix in Row Reduced Echelon Form? If not, explain why

[ 1 0 1 ]
[ 0 1 0 ]
[ 0 0 0 ]

yes

Is the following matrix in Row Reduced Echelon Form? If not, explain why

[ 0 1 0 ]
[ 1 0 0 ]
[ 0 0 1 ]

No because row one and two have leading ones that do not follow the step pattern

What elementary row operation is needed to turn the following matrix into Row Reduced Echelon Form?

[ 2 0 ]
[ 0 1 ]

1/2R1

What elementary row operation is needed to turn the following matrix into Row Reduced Echelon Form?

[ 0 1 0 ]
[ 1 0 0 ]
[ 0 0 1 ]

R2 ←> R1

What is an augmented matrix?

A matrix that includes both the coefficients on the right-hand side separated by a line

True or false that when turning and augmented matrix is turned into Row Reduced Echelon form, the constant side of the matrix must also follow the rules of Row Reduced Echelon Form

False

is the following augmented matrix in Row Reduced Echelon form? If not explain why:

[ 1 0 3 | 1 ]
[ 0 0 1 | 0 ]
[ 0 0 0 | 2 ]

No, because row three has a coefficient row of all zeros with a constant that is not zero

Is the following augmented matrix in Row Reduced Echelon form? If not, explain why:

[ 1 0 | 1 ]
[ 0 3 | 6 ]

no, because the first non-zero entry in R1 should be a leading 1

Find the value of K for which the following matrix is in Row Reduced Echelon Form:

[ 1 k ]
[ 0 1 ]

K must be 0

Find the value of K for which the following matrix is in Row Reduced Echelon Form:

[ 1 k 0 ]
[ 0 0 1 ]

K can be any number

Find the value of K for which the following matrix is in Row Reduced Echelon Form:

[ 1 0 0 2 ]
[ 0 1 0 0 ]
[ 0 0 0 k ]

K must be 0

What does “row equivalent” mean?

When a matrix is obtained from another matrix by means of elementary row operations

For the given augmented matrix, find one that is row equivalent in row reduced echelon form:

[ 2 0 | 1 ]
[ 0 1 | 0 ]

[ 1 0 | 1/2 ]
[ 0 1 | 0 ]