please help me For linear system x1 x2 2x3 2x5 3 2x1 2x

please help me

For linear system {-x_1 + x_2 - 2x_3 - 2x_5 = -3 -2x_1 + 2x_2 - 4x_3 + 4x_4 - 6x_5 - 2x_6 = -8 2x_1 - 2x_2 + 4x_3 - 2x_4 + 5x_5 + x_6 = 7 2x_1 - 2x_2 + 4x_3 + 4x_4 + 2x_5 = 6, write its coefficient matrix A, value vector b, and augmented matrix; perform and record a sequence of elementary row operations to find the reduced row echelon form of its augmented matrix; use definitions and (b) to determine the rank and nullity of matrix A; use (b) to determine whether the matrix equation Ax = b is consistent. If so, express its general solution in vector form first and then as a linear combination of specific vectors.

Solution

(a) The coefficient matrix for the given linear system of equations is A =

-1

1

-2

0

-2

0

-2

2

-4

4

-6

-2

2

-2

4

-2

5

1

2

-2

4

4

2

0

The value vector b =

-3

-8

7

6

The augmented matrix B =

-1

1

-2

0

-2

0

-3

-2

2

-4

4

-6

-2

8

2

-2

4

-2

5

1

7

2

-2

4

4

2

0

6

(b) We will reduce B to its RREF as under:

Then, the RREF of B is

1

-1

2

0

2

0

0

0

0

0

1

-1/2

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

(c) From the above, it is apparent that the RREf of A is

1

-1

2

0

2

0

0

0

0

1

-1/2

0

0

0

0

0

0

1

0

0

0

0

0

0

The rank of A is the number of non-zero rows in its RREF, i.e. 3

As per the Rank-Nullity theorem, the nullity of A is the number of columns in A –its rank = 3

(d) It is apparent from a scrutiny of the last row of the RREF of B that the given linear system is inconsistent ( as 0 cannot be equal to 1).

-1	1	-2	0	-2	0
-2	2	-4	4	-6	-2
2	-2	4	-2	5	1
2	-2	4	4	2	0