[PDF] Matrix multiplication

Let C = AB It is a m × p matrix Recall that the entry in the ith row and jth column of C, ie the (i,j)th entry of C, is called cij The entry cij is the product of the ith row

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[PDF] Matrix multiplication - The University of Sydney

Let C = AB It is a m × p matrix Recall that the entry in the ith row and jth column of C, ie the (i,j)th entry of C, is called cij The entry cij is the product of the ith row

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Matrixmultiplication

JackieNicholas

MathematicsLearning Centre

UniversityofSydney

c?2010Universit yofSydney

Multiplyingmatrices

WecanmultiplymatricesAandBtogethertofo rmthe product

ABprovidedthenumberof columnsinAequalsthenumberof

rowsinB. IfA= 4-13 1-29 andB= 0-5 -1-4 0-1 thenwe candefineABasAhasthreecolumns andBhasthree rows.

Multiplyingmatrices

IfA= 4-13 1-29 andB= 0-5 -1-4 thenABisnotdefined asAisa2 ×3matrix andBisa2 ×2 matrix;thenumber ofcolumnsof Adoesnotequal thenumberof rowsofB. Ontheother hand,the productBAisdefinedasthe numberof columnsofB,2,does equalthenumber ofro wsofA. Thistellsus somethingvery important;o rdermatters!! Inmostcas esAB?=BA.HereABisnotdefined whereasBAis.

Howtomultiplymatrices

Ingeneral,if Aisam×nmatrixandBisan×pmatrix,the productABwillbea m×pmatrix.

LetC=AB.Itis am×pmatrix.

Recallthatthe entryinthe ithrowandjthcolumnofC,iethe (i,j)thentryofC,iscalled c ij

Theentryc

ij isthep roductofthe ithrowofAandthejthcolumn ofBasfollo ws: c ij =a i1 b 1j +a i2 b 2j +a i3 b 3j +a i4 b 4j +···+a in b nj

Exampleofmultiplying matrices1

Thatprobably lookedabitcomplicateds owewillgothrough an example. LetA= 0-5 -1-4 6-2 andB= 0-1 -32

Aisa3 ×2matrixand Bisa2 ×2,so ABisdefined.

IfC=ABisthenC=

c 11 c 12 c 21
c 22
c 31
c 32
isa3 ×2matrix.

Exampleofmultiplying matrices2

0-5 -1-4 6-2 0-1 -32 c 11 c 12 c 21
c 22
c 31
c 32

Forthefirstentryc

11 wemultiplethefirstrowofAwiththefirst columnofBasfollows : 0-5 0. -3.

0×0+-5×-3.

iec 11 =a 11 ×b 11 +a 12 ×b 21
=15.

Exampleofmultiplying matrices3

0-5 -1-4 6-2 0-1 -32 15c 12 c 21
c 22
c 31
c 32

Fortheentryc

12 wemultiplethefirstrowofAwiththesecond columnofBasfollows : 0-5 .-1 .2 .0×-1+-5×2 iec 12 =a 11 ×b 12 +a 12 ×b 22
=-10.

Exampleofmultiplying matrices4

0-5 -1-4 6-2 0-1 -32 15-10 c 21
c 22
c 31
c 32

Fortheentryc

21
wemultiplethesecondrowofAwiththefirst columnofBasfollows : -1-4 0. -3. -1×0+-4×-3. iec 21
=a 21
×b 11 +a 22
×b 21
=12.

Exampleofmultiplying matrices5

So,tomultiply two matriceswe systematicallyworkouteach entryinthis way ,s tartingwiththefirstentry. 0-5 -1-4 6-2 0-1 -32 c 11 c 12 c 21
c 22
c 31
c 32
15-10 12-7 6-10

Clicktosee howw egetthe otherentries.

0-5 -1-4 6-2 0-1 -32 15-10 12-7 6-10

Exampleofmultiplying matrices5

So,tomultiply two matriceswe systematicallyworkouteach entryinthis way ,s tartingwiththefirstentry. 0-5 -1-4 6-2 0-1 -32 c 11 c 12 c 21
c 22
c 31
c 32
15-10 12-7 6-10

Clicktosee howw egetthe otherentries.

0-5 -1-4 6-2 0-1 -32 15-10 12-7 6-10

Exampleofmultiplying matrices5

So,tomultiply two matriceswe systematicallyworkouteach entryinthis way ,s tartingwiththefirstentry. 0-5 -1-4 6-2 0-1 -32 c 11 c 12 c 21
c 22
c 31
c 32
15-10 12-7 6-10

Clicktosee howw egetthe otherentries.

0-5 -1-4quotesdbs_dbs14.pdfusesText_20

[PDF] [PDF] Matrix multiplication - The University of Sydney

Matrixmultiplication

JackieNicholas

MathematicsLearning Centre

UniversityofSydney

Multiplyingmatrices

ABprovidedthenumberof columnsinAequalsthenumberof

Multiplyingmatrices

Howtomultiplymatrices

LetC=AB.Itis am×pmatrix.

Theentryc

Exampleofmultiplying matrices1

Aisa3 ×2matrixand Bisa2 ×2,so ABisdefined.

IfC=ABisthenC=

Exampleofmultiplying matrices2

Forthefirstentryc

0×0+-5×-3.

Exampleofmultiplying matrices3

Fortheentryc

Exampleofmultiplying matrices4

Fortheentryc

Exampleofmultiplying matrices5

Clicktosee howw egetthe otherentries.

Exampleofmultiplying matrices5

Clicktosee howw egetthe otherentries.

Exampleofmultiplying matrices5

Clicktosee howw egetthe otherentries.