-2 5. Given al-| 0 | , a2-1 3 |, ??- -2 6 | , and b- -3 -6 | . Determine if b ca
ID: 3185313 • Letter: #
Question
-2 5. Given al-| 0 | , a2-1 3 |, ??- -2 6 | , and b- -3 -6 | . Determine if b can be written as a 14 linear combination of a, az and a. That is, determine whether weights z1, 2 and z exist such that (a) (2pts) Write a matrix equation that is equivalent to the above vector equation (1). (b) (2pts) Write the augmented matrix of the above matrix equation (c)(7pts) Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form (d) (7pts) Decide whether the system has a solution. If it does have a solution, then continue row reduction to obtain the reduced echelon form of the augmented matrix and then find one solution for?1,22 and t3.Explanation / Answer
5. Let M = [a1,a2,a3,b] =
1
-2
-6
11
0
3
6
-6
1
-2
-3
14
The RREF of M is
1
0
0
9
0
1
0
-4
0
0
1
1
It implies that b = 9a1-4a2+a3.
a. The matrix equation equivalent to the given vector equation is AX = b, where X = (x1,x2,x3)T and A =
1
-2
-6
0
3
6
1
-2
-3
b. The augmented matrix is M, as mentioned above.
c. M can be reduced to its RREF by the following row-operations:
5. Add 6 times the 3rd row to the 1st row
6. . Add 2 times the 2nd row to the 1st row
The RREF of M is as mentioned above.
d. It is apparent from the RREF of M that the given system has a unique solution which is x1 = 9, x2 = -4 and x3 = 1.
1
-2
-6
11
0
3
6
-6
1
-2
-3
14
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