Linear algebra problem: Please show all steps and explain your solution, ensure

Question

Linear algebra problem:

Please show all steps and explain your solution, ensure that you are correct.

Question 9: We learned in the first part of the course that a plane in R3 can be described as the solution of a linear equation in the 3 unknowns (r, y,z), and that the intersection of planes can be described as the solution of the linear systems whose equations describe the planes that we want to intersect (see problem 5 of Test 1). Use this geometric interpretation of linear systems to cook up an example of a 3 × 3 system of linear equations that has no solutions. HINT: no hint

Explanation / Answer

9. Let the 3x3 linear system be 3x+2y+z = 3, x-3y+1z = 4 and 6x+4y+2z = -1.

The augmented matrix of this linear system is A =

3

2

1

3

1

-3

1

4

6

4

2

-1

To solve the3above linear system, we will reduce A to its RREf as under:

Multiply the 1st row by 1/3

Add -1 times the 1st row to the 2nd row

Add -6 times the 1st row to the 3rd row

Multiply the 2nd row by -3/11

Multiply the 3rd row by -1/7

Add 9/11 times the 3rd row to the 2nd row

Add -1 times the 3rd row to the 1st row

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

Then the RREF of A is

1

0

5/11

0

0

1

-2/11

0

0

0

0

1

A scrutiny of the last row of the RREF of A reveals that 0 = 1 which is not true. Hence the above system of linear equations is inconsistent, i.e. it has no solution.

3

2

1

3

1

-3

1

4

6

4

2

-1

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