SOLUTION: Let B =( matrix (3,3, 1,2,3, 2,k-3,3, 3,4,k-4 ) ) a)Find det(B) in terms of k; b)For what value(s) of k are the column vectors of B linearly dependent c)For k = 0, find

Algebra ->  College  -> Linear Algebra -> SOLUTION: Let B =( matrix (3,3, 1,2,3, 2,k-3,3, 3,4,k-4 ) ) a)Find det(B) in terms of k; b)For what value(s) of k are the column vectors of B linearly dependent c)For k = 0, find       Log On


   



Question 1006746: Let B =(
matrix (3,3,
1,2,3,
2,k-3,3,
3,4,k-4
)
)
a)Find det(B) in terms of k;
b)For what value(s) of k are the column vectors of B linearly dependent
c)For k = 0, find det(B) and det (-1/2B^tBB^-1) .

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
B+=%28matrix+%283%2C3%2C1%2C2%2C3%2C2%2Ck-3%2C3%2C3%2C4%2Ck-4%29%29

I'll do the first part, part (a), to get you started

I'm using the method shown below


(Image Source: Wikipedia)

Using that method, we can say



Let's compute the sub-determinants (the determinants of the 2x2 submatrices)

abs%28matrix+%282%2C2%2Ck-3%2C3%2C4%2Ck-4%29%29+=+%28k-3%29%28k-4%29-4%2A3

abs%28matrix+%282%2C2%2Ck-3%2C3%2C4%2Ck-4%29%29+=+k%5E2-7k%2B12-12

abs%28matrix+%282%2C2%2Ck-3%2C3%2C4%2Ck-4%29%29+=+k%5E2-7k
----------------------------------------------------
abs%28matrix+%282%2C2%2C2%2C3%2C3%2Ck-4%29%29=2%2A%28k-4%29-3%2A3

abs%28matrix+%282%2C2%2C2%2C3%2C3%2Ck-4%29%29=2k-8-9

abs%28matrix+%282%2C2%2C2%2C3%2C3%2Ck-4%29%29=2k-17
----------------------------------------------------
abs%28matrix+%282%2C2%2C2%2Ck-3%2C3%2C4%29%29=2%2A4-3%2A%28k-3%29

abs%28matrix+%282%2C2%2C2%2Ck-3%2C3%2C4%29%29=8-3k%2B9

abs%28matrix+%282%2C2%2C2%2Ck-3%2C3%2C4%29%29=-3k%2B17
----------------------------------------------------

Now let's return to the determinant of the 3x3 matrix.



det%28B%29+=+1%2A%28k%5E2-7k%29%2B%28-2%29%2A%282k-17%29%2B3%2A%28-3k%2B17%29 Make the proper substitutions (see the sub-determinants above)

det%28B%29+=+k%5E2-7k-4k%2B34-9k%2B51 Distribute

det%28B%29+=+k%5E2-20k%2B85 Combine like terms. This is the final answer for part (a)