rev matrices part ii
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7/21/2019 rev Matrices Part II
http://slidepdf.com/reader/full/rev-matrices-part-ii 1/25
Gdtrcmbs
Ebhrudry =, 8>42
7/21/2019 rev Matrices Part II
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^rapbrtcbs ae Gdtrcx Apbrdtca`s
(Gdtrcx Dllctca`)
4. + 7 +
8. + + 7 + +
=. \jbrb cs d u`cqub g x ` gdtrcx A sumj tjdt + 7 ear d`y g x ` gdtrcx D. \jb gdtrcx
A cs mdffbl g x ` zbra gdtrcx
<. Ear bdmj g x gdtrcx D, tjbrb cs d u`cqub gx ` gdtrcx (∘) s.t. + ∘ 7 . \jb
gdtrcx ∘ cs tjb `bidtcvb ae D.
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^rapbrtcbs ae Gdtrcx Apbrdtca`s
(Gdtrcx Guftcpfcmdtca`)
4. 7
8. + 7 +
=. + 7 +
7/21/2019 rev Matrices Part II
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^rapbrtcbs ae Gdtrcx Apbrdtca`s
(]mdfdr Guftcpfcmdtca`)
4. r 7
8. + 7 +
=. + 7 +
<. 7 7
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^rapbrtcbs ae Gdtrcx Apbrdtca`s
(\rd`spasb)
4. ()7
8. ( + )7 +
()
7
<. ()7
7/21/2019 rev Matrices Part II
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4. Clb`tcey d`l trd`searg gdtrcmbs c` raw
rblumbl bmjbfa` earg
8. Lbtbrgc`b bfbgb`tdry raw apbrdtca`s
=. ]afvb systbgs ae fc`bdr bqudtca` usc`i Iduss
” Oarld` rblumtca` pramblurb
Ahobmtcvbs?
7/21/2019 rev Matrices Part II
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Gdtrcx c` rblumbl bmjbfa` earg
Lbec`ctca`?
D` gdtrcx cs c` rblumbl raw bmjbfa` eargwjb` ct sdtcsecbs tjb eaffawc`i?
4. Dff raws ma`scstc`i b`tcrbfy ae zbrabs, ce d`y, drbdt tjb hattag ae tjb gdtrcx
> > >
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Gdtrcx c` rblumbl bmjbfa` earg
Lbec`ctca`?
D` gdtrcx cs c` rblumbl raw bmjbfa` earg
wjb` ct sdtcsecbs tjb eaffawc`i?4 . D ff r aw s ma s cs tc i b t cr bf y ae z br ab s, ce d y, dr bd t tj bh at ta g ae tj bg dt rc x
8. \jb fbdlc`i b`try ae bdmj raw cs 4
44
> > >
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Gdtrcx c` rblumbl bmjbfa` earg
Lbec`ctca`?
D` gdtrcx cs c` rblumbl raw bmjbfa` earg wjb` ctsdtcsecbs tjb eaffawc`i?4 . D ff r aw s ma s cs tc i b t cr bf y ae z br ab s, ce d y, dr bd t tj bh at ta g ae tj bg dt rc x
8. \jb fbdlc`i b`try ae bdmj raw cs 4
=. Ce raw d`l + 4 drb twa summbsscvb raws tjdt la `atma`scst b`tcrbfy ae zbrabs, tjb` tjb fbdlc`i b`try aeraw + 4 cs ta tjb fbet ae tjb fbdlc`i b`try ae raw
4
> 4
> > >
7/21/2019 rev Matrices Part II
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Gdtrcx c` rblumbl bmjbfa` earg
Lbec`ctca`?
D` gdtrcx cs c` rblumbl raw bmjbfa` earg wjb` ctsdtcsecbs tjb eaffawc`i?4 . D ff r aw s ma s cs tc i b t cr bf y ae z br ab s, ce d y, dr bd t tj bh at ta g ae tj bg dt rc x
8. \jb fbdlc`i b`try ae bdmj raw cs 4
=. Ce raw d`l +4 drbtwasummbsscvbraws tjdtla `atma`scstb`tcrbfy ae zbrabs,tjb`tjb fbdlc`i b tryae raw + 4 cs tatjb fbetae tjbfbdlc`i b`tryae raw
<. Ce d mafug` ma`tdc`s d fbdlc`i b`try ae sagb raw, tjb`dff atjbr b`trcbs c` tjdt mafug` drb zbra.
4 >
> 4
> > >
7/21/2019 rev Matrices Part II
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Bxdgpfb 4?
74 > >
> 4 >
> > 4
∘8
=
>
74 8 >
> > 4
> > >
>
>
4
74 > >
> 4 <
> > >
∘8
=
>
7
4 > > = >
> > 4 > >
> > > > 4> > > > >
Xblumbl raw bmjbfa` earg
7/21/2019 rev Matrices Part II
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Bxdgpfb 8?
74 8 >
> > >
> > 4
∘8
>
∘=
74 > =
> 8 ∘8
> > 4
<
2
8
74 > =
> 4 <
> 4 >
∘8
=
>
7
4 8 > = >
> 4 4 > >
> > 4 ∘8 4> > > > >
@at c` rblumbl raw bmjbfa` earg
7/21/2019 rev Matrices Part II
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Bfbgb`tdry Xaw Apbrdtca`s
4. C`tbrmjd`ib raws ae D
8. Guftcpfy raw r ae D hyd `a` zbra ma`std`t
=. Dll l tcgbs raw r ae Dta raw s ae D
↜
7
4 8 >
> > >> > 4
∘8
>∘=
74 8 >> > 4
> > >
∘8∘=
>
7/21/2019 rev Matrices Part II
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Bfbgb`tdry Xaw Apbrdtca`s
4. C`tbrmjd`ibraws ae D
8. Guftcpfy raw r ae D hy d `a` zbra ma`std`t
=. Dll l tcgbs raw r ae D ta raw s ae D
↜
4
8
+
7
4 8 >
> 8 8> < =
74 8 >
> 4 4
> < =
7/21/2019 rev Matrices Part II
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Bfbgb`tdry Xaw Apbrdtca`s
4. C`tbrmjd`ibraws ae D
8. Guftcpfy rawr ae D hy d a` zbra ma std`t
=. Dll l tcgbs raw r ae D ta raw s ae D ↜
∘8 +
7
4 8 >
> 4 4> < =
74 > ∘8
> 4 4
> < =
7/21/2019 rev Matrices Part II
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Lbec`ctca`? D` g x ` gdtrcx D cs sdcl ta hb raw
bqucvdfb`t ta d` g x ` gdtrcx H ce H md` hb
ahtdc`bl hy dppfyc`i d ec`ctb sbqub`mb aebfbgb`tdry raw apbrdtca`s ta D.
Bxdgpfb =?
7 4 8 < =8 4 = 8
4 ∘4 8 =
7 8 < : 54 ∘4 8 =
< ∘4 9 :
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Bxdgpfb =?
7
4 8 < =
8 4 = 84 ∘4 8 =
7
8 < : 5
4 ∘4 8 =< ∘4 9 :
]tbp =? 8 +
78 < : 5
4 ∘4 8 =
< ∘4 9 :
Jb`mb, gdtrcx D cs raw
bqucvdfb`t ta gdtrcx H
]tbp 8? ↜
78 < : 5
4 ∘4 8 =
8 4 = 8
]tbp 4? 8
7 8 < : 58 4 = 8
4 ∘4 8 =
⟿
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^ramblurb c` \rd`seargc`i d gdtrcx ta
cts Xaw Bmjbfa` Earg
4. Ec`l tjb ecrst (fbet ta rcijt) mafug` c` D `at dffae wjasb b`trcbs drb zbra. \jcs mafug` csmdffbl pcvatdf mafug`.
8. Clb`tcey tjb ecrst (tap ta hattag) `a`zbrab`try c` tjb pcvatdf mafug`. \jcs b`try csmdffbl tjb pcvat.
=. C`tbrmjd`ib, ce `bmbssdry, tjb ecrst raw wctjtjb raw wjbrb tjb pcvat ammurs sa tjdt tjbpcvat cs `aw c` tjb ecrst raw.
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<. Lcvclb tjb ecrst hy tjb pcvat sa tjdt tjb b`try c`tjb ecrst raw d`l pcvatdf mafug` cs `aw 4.
2. Dll guftcpfbs ae tjb ecrst raw ta dff atjbr raws ta
gdkb dff b`trcbs c` tjb pcvatdf mafug`, bxmbpt tjbb`try wjbrb tjb pcvat cs famdtbl, bqudf ta zbra.
5. Clb`tcey H ds tjb (g ” 4) x ` suh gdtrcx ae tjb
rbsuftc`i gdtrcx, ahtdc`bl hy lbfbtc`i gb`tdffy ctsecrst raw.
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9. Dll guftcpfbs ae tjb ecrst raw ae tjb rbsuftc`i
gdtrcx ta dff atjbr raws sa tjdt dff b`trcbs c` tjb
pcvatdf mafug`, bxmbpt ear tjb pcvat, hbmagb
zbra
:.Clb`tcey M ds tjb (g -8) x ` suh gdtrcx ae tjb
rbsuftc`i gdtrcx, ahtdc`bl hy lbfbtc`i gb`tdffy
tjb ecrst raw. Xbpbdt stbps 4-9 a` tjcs gdtrcx.
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Bxdgpfb <? Xblumb tjb eaffawc`i
gdtrcmbs c`ta cts raw bmjbfa` earg
4. 74 8 =
8 ∘4 4
= > 4
3
:
=
8. 7
4 4 8 ∘2 =
8 2 ∘4 ∘3 ∘=
8 4 ∘4 = ∘44
4 ∘= 8 9 ∘2
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+ + 8 ∘ 2 7 =
8 + 2 ∘ ∘ 3 7 ∘=
8 + ∘ + = 7 ∘44 ∘ = + 8 + 9 7 ∘2
+ 8 + = 7 3
8 ∘ + 7 : = ∘ 7 =
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Iduss-Oarld` rblumtca` gbtjal
^ramblurb?
4. Earg duigb`tbl gdtrcx
8. \rd`searg duigb`tbl gdtrcx ta rblumbl raw
bmjbfa` earg usc`i bfbgb`tdry rawapbrdtca`s
=. \jb fc`bdr systbg tjdt marrbspa`ls ta tjb
gdtrcx c` rblumbl raw bmjbfa` earg ahtdc`blerag stbp 8 jds bxdmtfy tjb sdgb safutca`s dstjb icvb` fc`bdr systbg.
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Bxdgpfb 2? ]afvb tjb fc`bdr systbg hy
Iduss Oarld` rblumtca` gbtjal
+ 7 >
8 ∘ + = 7=
∘ 8 ∘ 7 =
4 8 =
8 ∘4 4
= > ∘4
3
:
=
4 8 =
8 ∘4 4
= > ∘4
73
:
=
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]afvb tjb fc`bdr systbg hy Iduss
Oarld` rblumtca` gbtjal
4. + + 8 ∘ 2 7 =
8 + 2 ∘ ∘ 3 7 ∘=
8 + ∘ + = 7 ∘44 ∘ = + 8 + 9 7 ∘2
8. + + 7 > + 7 =
+ 7 4
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