Question

Calculate the determinant
\[\left[\begin{array}{ccccc} -2&5&0&-1&3\\ 1&0&3&7&-2\\ 3&-1 &0&5&-5\\ 2&6&-4&1&2\\ 0&-3&-1&2&3\end{array}\right]\]

Answer 1

i suggest you use microsoft excel function MDETERM() to compute the determinant

Answer 2

http://www.bluebit.gr/matrix-calculator/calculate.aspx
or u can use this website , and u get -1032

Answer 3

Well, I have to do all the calculations on paper so I can't just write the answer.

Answer 4

choose a row or column to break the determinent into the sum of 5 4x4 matrices (the one with the most zeros...does this method seem familiar from class?

Answer 5

one way is use elimination to translate to an upper triangular form. The determinant is then the product of the main diagonal.

I was able to simplify the first 2 columns before the numbers got quite large. But you can switch to co-factors for a 3x3 which is not too bad.

Answer 6

*remember a row swap multiplies the determinant by -1

Answer 7

Yeah, I know two methods of calculating the determinant of a matrix. One is using Gaussian elimination and the other method is the one you described.
phi, that's exactly what I did, but my answer was wrong. I did a little arithmetical error at the start. The reason I asked this question was to find out if there is a simpler/faster method of calculating determinant.

Answer 8

not 5x5 unfortunately, you gotta bang it out long form until you get them down to 3x3's

Answer 9

I don't know a particularly easy way... -1032 tells you you are dealing with ugly numbers.

Answer 10

I did 2 row swaps to get 1's in the pivot position
1 0 3 7 -2
3 -1 0 5 -5
-2 5 0 -1 3
2 6 -4 1 2
0 -3 -1 2 3

then reduced to
1 0 3 7 -2
0 -1 -9 -16 1
0 0 -39 -67 4
0 0 -64 -109 12
0 0 26 50 0

now I found the determinant of
-39 -67 4
-64 -109 12
26 50 0

using the bottom row (it has zero)
26*(-67*12 + 4*109) - 50*(-39*12 + 4*64)

multiply by 1 * -1 (which are on the diagonal)