For each matrix in problem 4 determine if the columns of the

For each matrix in problem 4, determine if the columns of the matrix span R3. Give reasons for your answers. (Make as few calculations as possible.)

Matrices are : it is hard to use row and column in here but, i will do the following.

Matrix A = [column 1 column 2] , where column 1 = (-3 2 4)

Column 2= (12 6 8)

Matrix B has three columns;[ column 1 column 2 column 3] where

Column 1 contains (3 -2 6)

Column 2 contains (7 -5 11)

Column 3 contains (1 3 -4)

where columns are vertical position in matrix.

Matrix C has four columns;

Column 1 ( -2 4 0)

Column 2 (3 0 0)

Column 3 (5 -5 0)

Column 4 (1 11 0)

Solution

1.We have A =

-3

12

2

6

4

8

We will reduce A to its RREF as under:

Multiply the 1st row by -1/3 ; Add -2 times the 1st row to the 2nd row

Add -4 times the 1st row to the 3rd row ; Multiply the 2nd row by 1/14

Add -24 times the 2nd row to the 3rd row; Add 4 times the 2nd row to the 1st row

Then theRREF of A is

1

0

0

1

0

0

Since neither of the columns in the RREF of A has 1 in the last row, the columns of A do not span R³.

2       B   =

3

7

1

-2

-5

3

6

11

-4

We will reduce B to its RREF as under:

Multiply the 1st row by 1/3; Add 2 times the 1st row to the 2nd row

Add -6 times the 1st row to the 3rd row; Multiply the 2nd row by -3

Add 3 times the 2nd row to the 3rd row ; Multiply the 3rd row by -1/39    

Add 11 times the 3rd row to the 2nd row ; Add -1/3 times the 3rd row to the 1st row

Add -7/3 times the 2nd row to the 1st row

Then the RREF of B is I_3. Hence the columns of B span R³.

3.   C =

-2

3

5

1

4

0

-5

11

0

0

0

0

We will reduce C to its RREF as under:

Multiply the 1st row by -1/2 ; Add -4 times the 1st row to the 2nd row

Multiply the 2nd row by 1/6 ; Add 3/2 times the 2nd row to the 1st row

Then the RREF of C is

1

0

-5/4

11/4

0

1

5/6

13/6

0

0

0

0

Since none of the columns in the RREF of C has 1 in the last row, the columns of C do not span R³.

-3	12
2	6
4	8