4.6. EXERCISES 125

7. Row reduce the following matrices to obtain the row reduced echelon form. List thepivot columns in the original matrix.

(a)

 1 2 0 3

2 1 2 2

1 1 0 3



(b)

1 2 3

2 1 −2

3 0 0

3 2 1



(c)

 1 2 1 3

−3 2 1 0

3 2 1 1

8. Find the rank and nullity of the following matrices. If the rank is r, identify r columns

in the original matrix which have the property that every other column may bewritten as a linear combination of these.

(a)

0 1 0 2 1 2 2

0 3 2 12 1 6 8

0 1 1 5 0 2 3

0 2 1 7 0 3 4



(b)

0 1 0 2 0 1 0

0 3 2 6 0 5 4

0 1 1 2 0 2 2

0 2 1 4 0 3 2



(c)

0 1 0 2 1 1 2

0 3 2 6 1 5 1

0 1 1 2 0 2 1

0 2 1 4 0 3 1

9. Find the rank of the following matrices. If the rank is r, identify r columns in the

original matrix which have the property that every other column may be writtenas a linear combination of these. Also find a basis for the row and column spaces ofthe matrices.

(a)

1 2 0

3 2 1

2 1 0

0 2 1



(b)

1 0 0

4 1 1

2 1 0

0 2 0

