i need a simple way to find the determinant of a 4X4 matrix?

Question

please follow this guideline
it must be comprehensible
keep it in lamence terms.
seriously simple

Answer 1

But but but but but....it isn't that simple. I guess the simplest way is a down-and-out flat-out correct formula, but the formula is gonna be messy and hard to memorize. Other than that, it is the expansion by minors process that your book is trying to explain to you.

Suppose matrix is

a b c d
e f g h
i j k l
m n o p

then the determinant is

afkp-aflo-ajgp+ ajho+angl-anhk-ebkp+ eblo+ejcp-ejdo-encl+ endk+ibgp-ibho-ifcp+ ifdo+inch-indg-mbgl+ mbhk+mfcl-mfdk- mjch+mjdg

Sorry, that's the way it is.

Answer 2

A computer might use a long and messy formula, but humans would do better to use expansion of minors. If you have a determinant:

a b c d
e f g h
i j k l
m n o p

To expand by minors, pick any row or column. We will pick the first row, and the above determinant becomes:

........f g h..........e g h..........e f h...........e f g
+a*..j k l......-b*..i k l......+c*..i j l.....-d*.. i j k
.......n o p.........m o p.........m n p..........m n o

Ignore the dots. They are just for spacing. It looks ok on my computer. I hope it does on yours also.

Note that you now have three 3x3 determinants. Note also the alternating plus and minus signs. In this case they are +-+-. This would be the case for any odd numbered row or column. If we had picked an even numbered row or column it would have been -+-+. Finally note that when I multiplied by a, which is in the first row and first column, those are the rows and columns excluded from the 3x3 determinant. When I multiplied by b, which is in the first row and second column, those are the rows and columns excluded from the 3x3 determinant. And so it goes. Eliminate the row and column from which your expansion term comes.

Next, you can expand each 3x3 matrix by minors in the same fashion, and you will get three 2x2 matrixes for each 3x3 that you expand. The 2x2 matrixes can be solved directly. For example:

a b
c d

= ad - bc

This is a general formula you can use on any 4x4 matrix. As a practical matter, various row and column manipulations are usually done to a matrix on a case by case basis, to make it easier to compute before going thru the procedure above. As a practical matter, the goal is to make three of the four coefficients of expansion above (i.e. a b c d) become zero before expanding. But that is another topic and not a requirement for solving.

Answer 3

There's a good description of a way to do it here:

http://mathforum.org/library/drmath/view/51968.html