Find the following determinants

[adc85]

I know how to find the determinant in general but these two problems here are tough for me:

1. Find the determinant of:
3 -1 0 0 0
-1 3 -1 0 0
0 -1 3 -1 0
0 0 -1 3 -1
0 0 0 -1 3

2. Find the determinant of:
0 2 2 2 2
2 0 2 2 2
2 2 0 2 2
2 2 2 0 2
2 2 2 2 0

Now, I know that I need to select a row or column that contains all zeros except for one number. The row or column that the one non-zero number is on would be selected as well. Then you would use the formula (for selecting row i and column j):

det A = (-1)^(i + j) * det A (without row i and column j)

But I can't figure out how to create a row or column full of zeros (except one element) for both of these problems. Thanks for any help.

 

[OlderDan]

I know how to find the determinant in general but these two problems here are tough for me:

2. Find the determinant of:
0 2 2 2 2
2 0 2 2 2
2 2 0 2 2
2 2 2 0 2
2 2 2 2 0

Haven't done one of these for an age. How about this one. Subtract the row below from each row

-2 2 0 0 0
0 -2 2 0 0
0 0 -2 2 0
0 0 0 -2 2
2 2 2 2 0

Add the first 4, then the next 3 then the next 2

-2 0 0 0 2
0 -2 0 0 2
0 0 -2 0 2
0 0 0 -2 2
2 2 2 2 0

Add the first 4 to the last

-2 0 0 0 2
0 -2 0 0 2
0 0 -2 0 2
0 0 0 -2 2
0 0 0 0 8

Subtrace 1/4 of last from each of the others

-2 0 0 0 0
0 -2 0 0 0
0 0 -2 0 0
0 0 0 -2 0
0 0 0 0 8

Det = 128

 

[CrusaderSean]

you can use cofactor formula or just do elimination like what OlderDan did. i prefer doing elimination and multiply diagonals to get determinant. i think that's how computer does it also. however on a test, you might have to write down the cofactors explicitly. i don't really remember all the details, but i believe the general idea is to reduce the problem from finding det of large matrix to a small one.
I'll illustrate w/ a smaller matrix
a b c
d e f
g h i

if you expand along the first row, it's
a * det(e f; h i) - b * det(d f; g i) + c * det(d e; g h)
so you can see, the general pattern is
(entry j from row 1) * (det of matrix w/ row 1, column j erased) * (-1)^(row+col)
i think the sign is built into cofactors, but from here i hope you can see its like reduction formula. i mean you can reduce it to finding det of 1x1 matrix.
det(e f; h i) = e * det(i) - f * det(h)
again the sign comes from (-1)^(row+col)