Suppose that a = V6 – 2 and b = 2V2 - V6 then
Answers
,
without calculating directly (that is without expanding along a column or row or
otherwise). Explain your answer.
det
1 3 3 2 ∈1
5 4 9 9 1
2 9 9 10
3 3 2 13 1
Solution: The determinant is 0, which can be verified in advance using your calculator.
To show that it’s 0 without expanding along columns or rows, you can do two things.
First, if you row reduce the matrix into row echelon form, you obtain the matrix
1 3 3 2 ∈1
0 1 13/3 12/3
0 0 1 18/25
0 0 0 0
.
This matrix has determinant 0 by multiplying along the diagonal (since it’s upper
triangular). Two matrices that are row equivalent do not necessarily have the same
determinant, but the determinant of one is a non-zero multiple of the determinant of
the other. Thus, our original determinant must also be 0.
Or, you can solve the problem this way. Use your calculator to find that the determinant is 0 in advance. Once you know that the determinant should be 0, that means
we should look for linear dependencies among the columns and among the rows. So
then we look for one, and we find that
Row 2 ∈ 2 · (Row 1) = Row 4.
Having a linear dependency among the rows then implies that the matrix has determinant 0, which is what we wanted to show.
(b) Let
v1 =
1
5
2
3
, v2 =
3
4
9
䎄 2
, v3 =
䎄 2
9
9
13
, v4 =
䎄 1
䎄 1
10
1
.
Are v1, v2, v3, v4 linearly independent or linearly dependent? Explain.
Solution: They must be linearly dependent. The matrix has determinant 0 (so is
singular). The rows and columns of such a matrix must be linearly dependent.