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John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

2x2, 3x3, 4x4, 5x5 matrix determinants

I thought I'd share this
interpretation of what
I've learned about
matrix determinants
from wikipedia.
I can't guarantee
it is perfect, but
I had fun making it.
It is drawn by hand,
so it might have mistakes,
and I don't know if I
understand determinants
perfectly yet.
Enjoy anyway.
Critique welcomed!

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igloo myrtilles fourmis

careless25

Real Member

Registered: 2008-07-24

Posts: 560

Re: 2x2, 3x3, 4x4, 5x5 matrix determinants

I haven't learned matrix determinants yet but it would be nice for some1 to explain to me what is going on.
still 15 and learning Math so i dont understand all of this advanced math.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: 2x2, 3x3, 4x4, 5x5 matrix determinants

A matrix is a bunch of numbers in a square or rectangle.
A determinant is an equation you form by multiplying
the red numbers and then subtracting the product of the black numbers.
I only have heard of square determinants so far.
So for 2x2 matrix, the determinant is by example:
3 4
9 2
determinant = 3*2 - 4*9, which is 6 - 36, or -30.
That is why I have 2 red X's
like this:
X
  X
and two black X's for the subtraction
like this:
  X
X
Then for bigger matrices, there are more things added and subtracted.
For 3x3, there are 3 products added and 3 products subtracted.
For 4x4, there are 12 products added and 12 products subtracted.
For 5x5, there are 60 products added and 60 products subtracted.
That's all I know so far, just learning...

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igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: 2x2, 3x3, 4x4, 5x5 matrix determinants

I've just learned there is something
called a minor matrix.  And it's neat
because you can use your knowledge
of determinants to create the minor
matrix.  And then, guess what?  You
can apply sign inversion to a checkerboard
pattern on the minor matrix to form
what is called the cofactor matrix.
Just make sure the upper-left entry
is not inverted, and create a checkerboard
from there, so everywhere the bishop chess
piece can go will not be inverted, which
leaves the other ones, that are inverted,
by multiplying by -1.  (I am only talking
about matrices with real numbers as
entries -- I don't fiddle with complex numbers
quite yet.)  Then after you have your
cofactor matrix, you can get another
matrix of the same size just by swapping
the (go left to right) columns with the
(go top to bottom) rows, and this matrix
is very useful apparently and goes by
many names, which causes confusion,
such as adjoint matrix, conjugate matrix,
and others with proper name "H****" something,
among others.

Last edited by John E. Franklin (2008-07-28 12:41:38)

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igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: 2x2, 3x3, 4x4, 5x5 matrix determinants

4x4 Matrix Talk...

The following
equations have not
been checked for
typos yet...

The plus and minus
signs are replaced
with P and M,
respectively for
googleable equations.
Lowercase letters
beside one another
are multiplied.

First we will name the 16 locations
of a "4x4 matrix" with letters.
(1st row with 4 locations)a b c d
(2nd row with 4 locations)e f g h
(3rd row with 4 locations)i j k l
(4th row with 4 locations)m n o p

Here is a "4x4 minor matrix"
created from above lettering.
(a location)
fkpPglnPhjoMfloMgjpMhkn
(b location)
ekpPglmPhioMeloMhkmMgip
(c location)
ejpPflmPhinMelnMfipMhjm
(d location)
ejoPfkmPginMeknMfioMgjm
(e location)
bkpPclnPdjoMbloMcjpMdkn
(f location)
akpPclmPdioMaloMcipMdkm
(g location)
ajpPblmPdinMalnMbipMdjm
(h location)
ajoPbkmPcinMaknMbioMcjm
(i location)
bgpPchnPdfoMbhoMcfpMdgn
(j location)
agpPchmPdeoMahoMcepMdgm
(k location)
afpPbhmPdenMahnMbepMdfm
(l location)
afoPbgmPcenMagnMbeoMcfm
(m location)
bglPchjPdfkMbhkMcflMdgj
(n location)
aglPchiPdekMahkMcelMdgi
(o location)
aflPbhiPdejMahjMbelMdfi
(p location)
afkPbgiPcejMagjMbekMcfi


Here is a "4x4 cofactor matrix"
using the above lettering.
(a location)
fkpPglnPhjoMfloMgjpMhkn
(b location)
eloPhkmPgipMekpMglmMhio
(c location)
ejpPflmPhinMelnMfipMhjm
(d location)
eknPfioPgjmMejoMfkmMgin
(e location)
bloPcjpPdknMbkpMclnMdjo
(f location)
akpPclmPdioMaloMcipMdkm
(g location)
alnPbipPdjmMajpMblmMdin
(h location)
ajoPbkmPcinMaknMbioMcjm
(i location)
bgpPchnPdfoMbhoMcfpMdgn
(j location)
ahoPcepPdgmMagpMchmMdeo
(k location)
afpPbhmPdenMahnMbepMdfm
(l location)
agnPbeoPcfmMafoMbgmMcen
(m location)
bhkPcflPdgjMbglMchjMdfk
(n location)
aglPchiPdekMahkMcelMdgi
(o location)
ahjPbelPdfiMaflMbhiMdej
(p location)
afkPbgiPcejMagjMbekMcfi

Here is the "4x4 adjoint matrix" using
the above a to p lettering.
(also called conjugate transpose, adjugate matrix, Hermitian...)
(a location)
fkpPglnPhjoMfloMgjpMhkn
(b location)
bloPcjpPdknMbkpMclnMdjo
(c location)
bgpPchnPdfoMbhoMcfpMdgn
(d location)
bhkPcflPdgjMbglMchjMdfk
(e location)
eloPhkmPgipMekpMglmMhio
(f location)
akpPclmPdioMaloMcipMdkm
(g location)
ahoPcepPdgmMagpMchmMdeo
(h location)
aglPchiPdekMahkMcelMdgi
(i location)
ejpPflmPhinMelnMfipMhjm
(j location)
alnPbipPdjmMajpMblmMdin
(k location)
afpPbhmPdenMahnMbepMdfm
(l location)
ahjPbelPdfiMaflMbhiMdej
(m location)
eknPfioPgjmMejoMfkmMgin
(n location)
ajoPbkmPcinMaknMbioMcjm
(o location)
agnPbeoPcfmMafoMbgmMcen
(p location)
afkPbgiPcejMagjMbekMcfi

Here is the "4x4 inverse matrix" of
a b c d
e f g h
i j k l
m n o p

I will use a capital D in place the
usual / sign used for division.
I will use a capital C in place of the left parenthesis ( .
I will use a capital J in place of the right parenthesis ) .

The equations are long, so I have broken or partitioned the
equation after every twentieth character, and the break to the next
line means nothing at all, just put it all back together to
understand it:
(a location)
CfkpPglnPhjoMfloMgjp
MhknJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(b location)
CbloPcjpPdknMbkpMcln
MdjoJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(c location)
CbgpPchnPdfoMbhoMcfp
MdgnJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(d location)
CbhkPcflPdgjMbglMchj
MdfkJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(e location)
CeloPhkmPgipMekpMglm
MhioJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(f location)
CakpPclmPdioMaloMcip
MdkmJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(g location)
CahoPcepPdgmMagpMchm
MdeoJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(h location)
CaglPchiPdekMahkMcel
MdgiJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(i location)
CejpPflmPhinMelnMfip
MhjmJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(j location)
CalnPbipPdjmMajpMblm
MdinJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(k location)
CafpPbhmPdenMahnMbep
MdfmJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(l location)
CahjPbelPdfiMaflMbhi
MdejJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(m location)
CeknPfioPgjmMejoMfkm
MginJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(n location)
CajoPbkmPcinMaknMbio
McjmJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(o location)
CagnPbeoPcfmMafoMbgm
McenJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ
(p location)
CafkPbgiPcejMagjMbek
McfiJDCafkpMafloMagj
pPaglnPahjoMahknMbek
pPbeloPbgipMbglmMbhi
oPbhkmPcejpMcelnMcfi
pPcflmPchinMchjmMdej
oPdeknPdfioMdfkmMdgi
nPdgjmJ

This is the end of the inverse matrix equation.
(4x4 matrices)

Last edited by John E. Franklin (2008-07-31 05:45:24)

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igloo myrtilles fourmis