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EPhillips1989

Member

Registered: 2007-11-03

Posts: 29

determinants

can anyone please help me understand how to compute the determinant of the following 4x4 matrix

Last edited by EPhillips1989 (2007-12-08 09:49:11)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: determinants

Unless the matrix is nice and you can use a special trick on it, finding determinants is an iterative process. You reduce a 4x4 determinant to some 3x3 determinants, then reduce those to some 2x2 determinants.

Here's how it works:

Each of the 3x3 determinants on the right hand side was made by looking at an element in the first row and multiplying it by the determinant made by ignoring the row and column that that element is in. You then alternate adding and subtracting them.

You find a 3x3 determinant by making it into three 2x2 determinants in the same fashion.
Luckily, those 3x3 determinants have quite a few zeroes in them, so that makes your job easier.

Also, you can focus on any row you like.
For example, by focussing on the third row instead, your matrix's determinant would be:

Notice that there are only three 3x3 determinants this time, because the first element of the third row is a 0 and so the associated determinant becomes 0. It's good to strategically pick rows with many zeroes to make your job a bit easier. One important point though: If you use an even-numbered row, then the alternating sequence of adding and subtracting starts with a minus.
To help you remember, you can think of a matrix like a chessboard, with a + in the top-left corner.

Some special matrices let you find their determinants straight away.

For example:

If any size of matrix has a row made entirely of zeroes, then you can instantly tell that its determinant is 0. It can be any row of the matrix. (x denotes an unimportant element)

If any size of matrix has all zeroes above and to the right of the main diagonal (the line made of a's) then its determinant is the product of the elements of the main diagonal.

The same is true if the matrix has all zeroes below and left of the main diagonal instead.

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It wanted to be normal.

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: determinants

as an addition, you can instantly tell a determinant is 0 if any collumn is made of zeroes aswell, since you can also expand the determinant along a collumn in the same way you expand along a row

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The Beginning Of All Things To End.
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Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: determinants

Perhaps this would be a good idea for a flash script.  Write a flash app which not only takes the determinate of an nxn matrix, but also describes each step as it does so.  It's a good use of recursion.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Kargoneth

Member

Registered: 2007-08-11

Posts: 33

Re: determinants

Determinants... such interesting constructions.

A matrix's determinant is also zero if:
- Two rows have the same entries
- Two columns have the same entries

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: determinants

True. That's because you can add or subtract one row of a matrix to/from another without changing its determinant. If you subtract a row of a matrix from another equal row, then the result would be a zero row, and so you can use the previous result.

Other rules:

- Swapping two rows of a matrix causes the determinant to change sign.

- Multiplying a row by a scalar makes the determinant multiply by the same scalar.

- If you replaced all uses of "row" in this post with "column", then everything would still be true.

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Why did the vector cross the road?
It wanted to be normal.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: determinants

I've just learned the recursive "laplace expansion" for successively larger square determinants.
It looks as though this post matches what I learned in wikipedia.
I have drawn a pictoral depictment of
the 2x2, 3x3, and 4x4 sum of product terms,
where red is positive and black is a negative term.
Click on photo, examine, and let me
know if it looks right.  Thanks guys/gals.

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igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: determinants

Determinant answer appears to be 1, just positive 1,
providing my funny red/black blocks are correct.

These are the 24 terms in the order of my black/red dots.
0+0+0- -6    +0+0+0+0  +0+0+0+0    0+4- -6-18   +3+0+0+0   +0+0+0+0

This determinant applet I found!! Goes up to 5 by 5, wow.
http://www.fyzika.sk/java/matrix1.html
It agrees with 1 for answer.

Last edited by John E. Franklin (2008-07-25 08:45:02)

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igloo myrtilles fourmis

graciela

Member

Registered: 2010-08-14

Posts: 1

Re: determinants

3 1 1 1
1 3 1 1
1 1 3 1
1 1 1 3

Can someone explain how to solve this 4X4 Matrix! I need to find the determinant!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: determinants

Hi graciela;

Welcome to the forum!

The determinant = 48.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.