can anyone help with this determinants problem?

Question

okay so im sure no one will help, it is a long problem. BUT just in case.

how can i solve this determinants problem:

2 32 1 4
26 104 26 -13
2 56 2 7
1 40 1 5

the answer is 45. I have no clue where to begin!! if anyone can at least help me start i would be more than grateful


THANKS!!!!!!!!!!!

sorry but the spacing doesn't show well
it is a 4x4 determinant tho

ims upposed to solve this by first changing it to upper triangular....

Answer 1

Well, the fastest way to do this is to change this to upper-triangular form and then multiply the elements along the diagonal. Thus:

[2 32 1 4]
[26 104 26 -13] -13 (I)
[2 56 2 7] - (I)
[1 40 1 5] -1/2 (I)

[2 32 1 4 ]
[0 -312 13 -65]
[0 24 1 3] +1/13 (II)
[0 24 1/2 3] +1/13 (II)

[2 32 1 4]
[0 -312 13 -65]
[0 0 2 -2]
[0 0 3/2 -2] -3/4 (III)

[2 32 1 4]
[0 -312 13 -65]
[0 0 2 -2]
[0 0 0 -1/2]

The diagonal elements are then 2, -312, 2, -1/2. Their product, and the determinant of your matrix, is 624. Btw, I have no idea how you got 45. I put the matrix into a computer algebra system to check my work, so the determinant IS 624 - either you typed the matrix in wrong or your answer key is screwed up.

Answer 2

2 32 1 4
26 104 26 -13
2 56 2 7
1 40 1 5
c1->c1-c3

1 32 1 4
0 104 26 -13
0 56 2 7
0 40 1 5
r1-> r1-r4

1 -8 0 -1
0 104 26 -13
0 56 2 7
0 40 1 5

r2->r2-r3
1 -8 0 -1
0 48 24 -20
0 56 2 7
0 40 1 5

c4->c4+c1
1 -8 0 0
0 48 24 -20
0 56 2 7
0 40 1 5

c3->c3+c4
1 -8 0 0
0 48 4 -20
0 56 9 7
0 40 6 5
c2->c2/8
1 -1 0 0
0 6 4 -20
0 7 9 7
0 5 6 5

c2->c2+c1
1 0 0 0
0 6 4 -20
0 7 9 7
0 5 6 5

i think this will help u out upto great extant to solve further
here c reresents column
r represents row

Answer 3

check this site out! it will help a lot!
http://hyperphysics.phy-astr.gsu.edu/hbase/deter.html
the example on the second half of the site will show you the way