please help me For linear system {-x_1 + x_2 - 2x_3 - 2x_5 = -3 -2x_1 + 2x_2 - 4
ID: 3034650 • Letter: P
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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.Explanation / Answer
(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
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