Calculating the Determinant of a 4x4 Matrix

[John777]

Prove determinate of matrix is 150

1 2 3 4
4 3 2 1
0 -1 2 3
1 6 4 -2



My solution is incorrect and maybe my method is incorrect but I can't figure it out. I went down the first column to make use of that zero.

Here is the basic method (a_ij)(-1)^ij*Det(3x3) >> (a_ij)(-1)^ij * (a_ij)(-1)^ij*Det(2x2) where the second a_ij and (-1)^ij are for the newly created 3x3 matrix

First term:

(1)(-1)²(3)(-1)²[(2)(-2)-(4)(3)] = -48

Second term:

(4)(-1)³(2)(-1)²[(2)(-2)-(4)(3)] = 128

Third term = 0

Fourth term

(1)(-1)⁵(2)(-1)²[(2)(3)-(2)(1)] = -8

Det = -48 +128 - 8 = 72.

 

[Mark44]

I get 150. Check your work.

 

[John777]

I get 150. Check your work.

Well I knew it was 150, unless the textbook was incorrect. Does my method seem accurate?

 

[Mark44]

No, but I'm not sure I understood what you were trying to say. You are expanding down the first column, which means you will be evaluating three 3x3 determinants.

You can make life a little easier by subtracting the first row from the fourth row. That way, you'll need to evaluate only two 3 x 3 determinants. It's possible there are some other row operations you can do to simplify things before you start evaluating things.

 

[John777]

Here is what I did for the first part of the first column:


(1)(-1)² * det of
3 2 1
-1 2 3
6 4 -2

This simplies to

(1)(-1)²(3)(-1)²* det of

2 3
4 -2

which simplifies to
(1)(-1)2(3)(-1)2[(2)(-2)-(4)(3)] = -48


Sorry I wrote the matrices on the next line. I hope that's not to confusing but I didnt know how to type it otherwise.


Is this correct math?

 

[Mark44]

Here is what I did for the first part of the first column:


(1)(-1)² * det of
3 2 1
-1 2 3
6 4 -2

OK to here.

This simplies to

(1)(-1)²(3)(-1)²* det of

2 3
4 -2

No. To evaluate a 3 x 3 determinant, you need to evaluate three 2 x 2 determinants, not just one. I think that this is the mistake you're making.

which simplifies to
(1)(-1)2(3)(-1)2[(2)(-2)-(4)(3)] = -48


Sorry I wrote the matrices on the next line. I hope that's not to confusing but I didnt know how to type it otherwise.


Is this correct math?

 

[John777]

OK to here.
No. To evaluate a 3 x 3 determinant, you need to evaluate three 2 x 2 determinants, not just one. I think that this is the mistake you're making.

Oh your correct. That's stupid on my part. Had someone just put a 3x3 in front of me I wouldn't have had a problem. Just all together I got screwed up. Thanks.