SOLUTION: Determine a value for x such that the following vectors are linearly independent (1; 1; 2; 1); (2; 1; 2; 3) (1; 4; 2; 1); (-1; 3; 5; x)

Question 1094594: Determine a value for x such that the following vectors are linearly independent (1; 1; 2; 1); (2; 1; 2; 3)
(1; 4; 2; 1); (-1; 3; 5; x)
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
the 4 given vectors in R4 are linearly independent if and only if the determinant of the matrix formed by taking the vectors as its columns is non-zero.
:
the matrix A is
:
| 1 2 1 -1 |
| 1 1 4 3 |
| 2 2 2 5 |
| 1 3 1 x |
:
the det of A is expanded into four 3 by 3 determinants
:
1 times
| 1 4 3 |
| 2 2 5 | = 1*(2x-5) -4*(2x-15) +3*(2-6)
| 3 1 x |
:
-2 times
| 1 4 3 |
| 2 2 5 | = -2 * {1*(2x-5) -4*(2x-5) +3*(2-2)}
| 1 1 x |
:
1 times
| 1 1 3 |
| 2 2 5 | = 1*(2x-15) -1*(2x-5) +3*(6-2)
| 1 3 x |
:
1 times
| 1 1 4 |
| 2 2 2 | = 1*(2-6) -1*(2-2) +4(6-2) = 12
| 1 3 1 |
:
the sum is
:
1*(2x-5) -4*(2x-15) -12 -2(2x-5) +8*(2x-5) +(2x-15) -(2x-5) +12 +12 =
:
6*(2x-5) -3*(2x-15) +12 =
:
12x-30 -6x+45 +12 =
:
6x +27
:
6x +27 is not = 0 for all x EXCEPT -27/6 = -9/2 = -4.5
:

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