


Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
A comprehensive overview of matrix operations, including addition, subtraction, multiplication, and determinant calculation. It also covers basic concepts in linear algebra, such as vector spaces and linear transformations. Numerous examples and exercises to illustrate the concepts presented.
Typology: Study notes
1 / 4
This page cannot be seen from the preview
Don't miss anything!



[ ^]]E`&a]Ebc@dec@]"f g
h (i<#-!j#025,-108k5^ lE<B=mn-5oqp"r0st#5ou-qvw25^6 lsA
hyx 2 !jz)#5ou0-@v?25761ls^ (T#{<Bmn-15^ 6)|}#l~
h r=66X-S1-7w )<5^5 oOA
h k5o^5 o&r=5^ -B5qS6'-:1!j-'-5o#/B^5 o&6161!j#^5 o#-:!C6:ln25q-1B!$6vB026ls"#l"-B!$6<5o5oO^6 >Cr/!C6:l25@#5ou0- 5 7ln#5 6 lsA
\tg
5 q6'B>j>).5 ,>$!-1-1>$5qr2n-:1Ssn6?!$#/3>C66Vr2n-: 6X-O!j-!$65o0!$5oy#l&T - B615#-V+ !j>$!$!j-!$6l!$6326615 l !$#-B5}rOA vB5o5)15}&T5om!$#-S6-1 5o 5 o&r=5
h *@573>C5,#E1SsO^ P7nN0:
h <157.-15)#_1SsO^ N0¢¡£W¤¥P7n4¦K
h 5 o#-1B~Ss^ P70§Q£¡N0R1QWn¨0©<
h *@qª)«,v{2n-Ur2#=l6o^ N0'¬¦ ~N0¦K0K ~N04¦K >$>.-B1.®^ N¯O°7nYnW±²Z0²³´μ.Z0nYI.¶±H.W·n=Pμ A
h vB5@Ss¸.!$r>$5}¹1º+»ºo¹S¼^ - 1¸-1B5,sFrn½X5^3 -^ A %-<!C6G¾¿.À-1B5,3-12>=SsAVnÁ^ Pn£NJáN: !C6#- "3om0sÄr2n-@-1B5}5oun34-q6 5 }sA
hyŸÆ=ÇÉÈ +ÊVË_#ESs/3o#~r5Fln#5)r0s/Ä0!$#Ä"12n-1!$#5)-1Äln^ !j-q261!$#"-1B5F3o>j#5}^5 -1B0lE6T>$>j.q6^ Pn£NÌáÎÍ PnÂÏN0RSHQμW ÍÏU ªq-!$3o5@-B5,-Xs0m=5o93 6'-1!$#"l#5}.--1B5}r=5 !$##!$#AvB5}3>$#5G 5 -Bnl~15o-12#6F-Xs0m5¸«Grn½X5 3 -61 !-¸2=6X-r 3 o6'--1"-1B5}m1m=5 -Xs0m=5Á!j#~-B!$63 615}#EsA
h 66'!$#sn6,!C6}-1B5t61^5 t6}m=66'!$#-B5o^ «Grn½X5^3 -6^ AEvB5t3#-5o#-6&3o#Ðr5tÄ#!jm2>$-157lÑ2#>$!j5/!-B #Ä>Om!j !j-1!$5Fl-&-Xs0m=576oA
Ò Ó£ÔogUfÕcqFÖy\Ø×ÚÙÎÛ ÜU]"f \tÝÞ}ßáàâÖo}gwg
5 G5q#.Ãr=2n--1"6X-2lns 2S6X-<157>=D@r6'-1S34-*G-¸v<s0m=5Á-B5@%Â#-SDqsnã¸<>C66oA D@6<!-S6:#= 5 @6125 6 '-6 Á
h <#6'-1234-S6o(å-X^ w-1B5^ Ä ·Z0è=QPH£° n 0nN¯´N¨Í P7ÉP OPN0Qé0N·NnHnP7¯Ï 9 :2615^6 !j5o#~!j#!j-1!C>U3^ m3!j-Xs/V-1B5}r ·Z0è=QPH£° n 0nN¯´N¨Í Ï^9 2615^6 3^ m3!j-Xs/^6 '!$zo5^ |7
h llÉ^2 615^ l-^ llE#~5o>$5o^5 o#-¸( .--1B5}5o#lV-1B5}r= ·Z0è=QPHëênμ=P.Y¢NY0YÍoP7n£WnQWMOWnÏ
h15^ ".55o^ .5"/m'-!$3o2>C,!-5oìTÕ-B5^ r=A}%-1B5Ñí'-5o-îÄ.>j25^ !j5o#^ !C6,m1576'5^ #-,!j-,!C6)5o^ .57lÉA ªq-5,-1B!C6!C6-1B5F34-2>U3o#-15 #-?-1B5}rÄ#=l®¾O¿.ÀÉ!j-6>$n3o-1!$#UA ·Z0è=QPHïèμ0μnQW0NÃWMOμêW ÍP70ðN0¨nWOÏ 9 <Bsln05^6 -1B!$6#5)15o-12#~r=0>$5^ #=ñ
h^6 '!$zo5^6 1!$zo5)V-1B5}r=_(T#02&r=5^ ^5 o>$5o^5 o#-6S ·Z0è=QPH£P nòInP.ónW ÍoÏ
h^3 2#-«,3o3o215o#=35^6 <2#--1B5}#02&r=5^ w-1!$"576q^ !$5^ #í'-5o-î"mm5^ 6!$#~-1B5}r ·Z0è=QPH£P nòHμZ0n²H0HZnWOHWIUÍ P7ëN0¨WOÏ
h llD@>j>iD@ll6-1B5F3#-5o#-6?-1B5}!$5o#~r&-"-1B5tôoõn¹4¹1ºo¾ÀÉrA*@0576#-@ç5734-@3#-15o#-S6w!j5o#~r=A ·Z0è=QPHëênμ=P.Y¢NY0Y Q0QVÍ °70 0N¯´N¨öNYYnWYÏ
h^2 #!$#U0!$^ !j>C-1ÄllDq>$>Ér2n-q3o157.-1576t¾Oº÷®r&T1â-X<"!$5^ #tr6^ A %-l576#-q.çO5^3 -5o!j-1B5^ r=_ø.¾ù !$6 (^) I.NPH A Bsñ ·Z0è=QPHÃINOPH£° n 0nN¯´N¨Z0ÉPμ:Í °7n 0N¯´N¨ïè?¦ÉúѰ7n 0N¯´N¨öèJ=Ï
h^3 >$#5^ æ{576"-125¸«,ûòV-1B5}r=A ·Z0è=QPHy².èü0WH.§HQμW ÍoÏ
h^3 om=3!j-1!$5^6 æ_5-Bnl6-1/l5^ >É!j-1BEB.¸23SBE61m3o5G-B5}r"!C6q34-12=>$>js/2=6'!$# ·Z0è=QPH£P n£¨WnénN·NHnP¯Í Ï^9 5-121#63^ m3!j-Xs ·Z0è=QPHëênμ=P.Y¢WOIZ0WénN·NHnP0¯ÍoP7n®MwP ÉPMZM=énN·NH0P0¯OÏ 9 )^ 5~6'25/!j-"B=6&5o#2B61m35ÄT&^ !$#! ¸2 ·Z0è=QPHëênμ=P.YÃ0OPMýμ0þ=P.ó0W ÍÏ^9 -1!$Ä6)3om=3!j-Xs_ln.#^ - t-125¸#02¸r5o):5o>$5o^5 o#-S6G!$#Ðr=A)æ^ 6'-)61m3o 5 ÿ/3!$5o#- A
vB5Är0 576)!j#-_"15Äln5o-!j>#Ñ5 3SBÐ 5 -BnlÉÁwB=.-}-1B5Äm15o9 .m=6X-193o#l!-!j#6)5&T}-1B5 6 15Ä"5o-1Bnl #=l®B=.-F5u3o5omn-!j#6}5"-1B.#UÁ?!$#3o>j2l!j# #«G2n-4«@+æ5 s:17A Dq>j-1B2BÑ5/6'B>j>#-&ln57>!j-1BÑ-B!$ !C616125!jB-#.}Á 5 o s&>j>$n3o.-!j#&!C63o5o1-!$#>$s¸#5O5 - !$ m='-S#3o5A5os}-1!$ 5 qs2"3o157.-15q)#5oyrn½X5734- Á 2 61!j# (^) W¤ Á7-1B5 ½'.G!j#-15 1m15o-15o"õ0¼4À=#lF-B.- 5 o 1sF61 5 oB5o5A?%-2615 6 B=.-!$6 3 >$>j57lF-B5m1SE 6 w (^) Ç Á B!C3SBï3#-S!$#6}>$> 5 o s{.!$>Cr>$5"-1E-1B5/mSEA B5 #Ñ-1B!C6} 5 o s !C6}5unB26'-15 lUÁ?S5"r2n-¸!$#Ð>C5 m 6 :3o15 >j57616<m 6 <!j-B=mm5o#6 A$AjAs2 VÊ (^) Æ w (^) Æ ëÆ È AwD--1B!C6m!$#-7Án#5) Ê m=5 -1!$# !C6}3o>$>$5 lÉÁU#®5oun3o5omn-!j#{!C6,-1B.#®#l{-1B5 m 5 !-B5o)-5o"!$#-15 6 ,G-B5ÄmÎ3 .-3SB5 6 G-1B5 5 oun3o5omn-!j# #=lln57>C6!-B!-q35oT2>j>$>$sA
! "$#
vB!C6m3S.5,!j m>$5o 5 o#-S6-1B5)r"2=6'!$# #Es V!j#-5o5 6)(+B5o#3o5,-1B5}# 5 784A;-1B615G-B.-ÄsÄ5 3 o>j>É!j- Á !j-<!$>j>2615G}m'-!$>j>$s9 =>j>$5 lÄ1SsAVvB.-!$6 Á-B5G1Ss¸!$>$>=r5Án6sÁ61!$zo5F|7Ár2n--1B5 15GÄs&#>js"r5¸| F5o>$5o 5 o#- !$#E-1B5 15AvB!C6!$#0>$5765 5 om!$# - 1S3S % (^) Æ .!Cr>$5 6 A<vB5&s/#l~-1B5¸#2¸r5oq 5 >j5 5 o#-6!$#E-1B5¸sA D@3-12>j>$sÁOs2{>$61/#5 5 l_-t0#.-1B5"61!jz 5 ¸-1B5"sÁr2-GT'-2#.-5o>$sÁ-1B!$6,!C6G6'-1157lErs~-B5"1Sst!j-615o>jXA)%Â# &'&&Á0s2~<2>$l~>C61"#5o5 l-"5o5 mt-3SÄV-1B!C6.>$25A vB5qm=5 -1!$#615)(2!j-15G6X-!jB-1Tl¸5u35 mn-:B5 # -1B5,s}5-S6T2>$> Aª@.}Álln!j#¸# 5 o>$5o 5 #- 5 #= 5 6 1!jz !j#¸-1B5)SsÁB!$3SB/!$#0>j5 6 :>j>$n3o-1!$#&¸#5o¢1Ss"U>C5o6'!$zo5)#=l/3ms0!$#F-B5,!j#nT1Ä-1!$#ÄTâ#
·0Z0èQPHy²èüWH.§HQμW ÍÏ,+ °7n N¯´N¨ÃNOI7¤W:
00¯-+ NI¤W£¡ÎÍ °7n 0N¯´N¨OÏëIZ0·W<RSHQμW ÍÏ?
. HNH©ÎÍéQμW0/μnþZ0·0·μ0¶0±HW·OP.μyW=Ï1+ ©0줣W¤ 32 Z00OPMOW¶±HW·nPμ
ay{zmX%fe¢u^a%]^akGVL\^]¬)5u^a%]^aexs%WSl$\mXQrO^X%]%v^%\ay\{zmX%
qSvS}~W~VysVL^]s%l$vS^]m©^sSsv^S^Xy{syXmz VL^]o]^a%Sd^X%]5U VL\^]kay\{zmX% VL^]oVL^u^X% jmly|VL\^u^X% 'VL\^u^X%[n5ay\^b5Z]^XnOz0VL\^u^X%S¡S¡5 Vj>]^a%Sd^X%]kSu^a%]^aVL^u^X%S5 ay{zmX%S¡S¡^X%]%v^%\ay\{zmX%
qSvS}~W~Vysiml~Vu Xy{zv^^XSSayqmamsmV]Sb0VL\^]kn0VL\ V>n~v$n5SayqmamsmV]Sb5 VL\^]} VdSd^X%S_SS^a%bfC
Vj|>u^a%]^aeW%Xy^dS]%p[n0VL\ V>n~v$n5SayqmamsmV]Sb5 }VdSd^X%S_SS^a%bk\mXy VL\^]f¥n0VL\ V>n~v$n5SayqmamsmV]SbS ¦b~z$]^Xn6eha%S^a%b~s%l$q^b0>u^a%]^a§§} VdSd^X%S_SS^a%b§'§n5ay^b5Z]^XnOzmO u^a%]^a} VdSd^X%S_SS`^a%b
qSvS}~W~VysVL^]d^X%]^SayqmamsmV]Sb0^X%]%v^%\ku^a%]^aeW%Xy^dS]%pf'
qSvS}~W~Vysr}~lSlSW%XSay\k^Xn lyi^XQVL\^]]^a%Sd^X%]5r VL^]oVL^u^X%¨RSRr%pmX WSl^sya%]5V$l$\lyj]^a%Sd^X%]oVL\k]%pmXu^a%]^a a%S^a%bfe jmly|VL^u^X% UVL^u^X%[n5ay^b5Z]^XnOzmU®S®|>]^a%`Sd^X%]:¯°u^a%]^aVL^u^X%S5OUVL^u^X%S¡S¡5O
Vj|VL^u^X% Sn5ay^b5Z]^XnOzm ^X%]%v^%\kj^a^W^zyX X^W^zyX n5ay^b5Z]^XnOz%±S± u^a%]^aVL^u^X%Sku^a%]^a¥n5ay^b5Z]^XnOz$ ^X%]%v^%\k]S`%vmX
qSvS}~W~VysVL^]zmV²^XQU^X%]%v^%\n5ay^b5Z]^XnOz0¨
qSvS}~W~Vysiml~Vu]S5V>n{ml ¦V²^XQ VL\^] ]S5V>nSn5X%uS_SS^a%bfC
Vj|>u^a%]^aeW%Xy^dS]%p¯°n5ay^b5Z]^XnOzm ]S5V>nSn5X%uS_SS^a%bk\mXy VL\^]f¥n5ay\^b5Z]^XnOz$ ¦b~z$]^Xn6eha%S^a%b~s%l$q^b0>u^a%]^a§§)]S5V>nSn5X%uS_SS^a%b§'§n5ay^b5Z]^XnOzmO u^a%]^a]S5V>nSn5X%uS_SS`^a%b
qSvS}~W~Vyskz$]^a%]5VysZL^]S_SS^a%bSc^a%dkvS\ V$l$\ZL^]S_SS^a%bSc^a%d} O§[ZL^]S_SS^a%bSc^a%dk}m¤{ ZL^]S_SS^a%bSc^a%day{zmX%\mXyoZL\^]S_SS^a%bSc^a%d0x} Qe¢d^X%]^SayqmamsmV]Sb0[¡}m¤e¢d^X%]^SayqmamsmV]Sb0 SO ¦b~z$]^Xn6eha%S^a%b~s%l$q^b0x} Qe¢u^a%]^af§§Bay\{zmX%fe¢u^a%]^a§'§} Qe¥n5ay^b5Z]^XnOzmO ¦b~z$]^Xn6eha%S^a%b~s%l$q^b0x}m¤e¢u^a%]^af§§Bay{zmX%fe¢u^a%]^a§¨} Qe¥n5ay\^b5Z]^XnOz0§¨}m¤e¥n5ay\^b5Z]^XnOzmO ay\{zmX%fe¥n5ay^b5Z]^XnOzkr} Qe¥n5ay^b5Z]^XnOz ¡r}m¤e¥n5ay^b5Z]^XnOz0 ^X%]%v^%\ay{zmX%`
³