Matrices and Determinants

Matrices

1. UNIT 4
Matrices and
Determinants
Mr.T.SOMASUNDARAM
ASSISTANT PROFESSOR
DEPARTMENT OF
MANAGEMENT
KRISTU JAYANTI COLLEGE,
BANGALORE

2. UNIT 4
MATRICES AND DETERMINANTS
Introduction,Meaning,TypesofMatrices,
Operationsofaddition,subtractionand
Multiplicationoftwomatrices,Problems,
Transposeofsquarematrix,Determinantsof
squarematrix,minorofanelement,cofactorof
anelementofdeterminant,Adjointofasquare
matrix,singularandnon–singularmatrices,
Inverseofasquarematrixsolution,problems
onlinearequations,twovariablesusing
Cramer’srule.

3. MATRICES -INTRODUCTION
Definition:
“Anarrangement(array)ofnumbers(realorcomplex)
intheformofrowsandcolumnswithinabracketiscalleda
Matrix.”
“Amatrixisasetorgroupofnumbersarrangedina
squareorrectangulararrayenclosedbytwobrackets”.
ThenumbersthatformamatrixarecalledElementsofthe
matrix.
ThematricesaredenotedbycapitalletterA,B,C,……
IfmatrixAhasmrowsandncolumns,thenAiscalledthe
matrixofordermXn.

4. Properties:
•Aspecifiednumberofrowsandaspecifiednumberof
columns.
•Twonumbers(rowsxcolumns)describethedimensionsor
sizeofthematrix.
Matrixalgebrahasatleasttwoadvantages:
•Reducescomplicatedsystemsofequationstosimple
expressions
•Adaptabletosystematicmethodofmathematicaltreatment
andwellsuitedtocomputers.
(1x2) (2x2) (2x2)11 
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

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03
24 
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dc
ba

5. (E.g.)
is a matrix of order 3 x 3
is a matrix of order 2 x 4
is a matrix of order 1 x 2


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




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
333
514
421 



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 
2
3
3
3
0
1
0
1 11

6. Matrices -Introduction
Amatrixisdenotedbyaboldcapitalletterandtheelements
withinthematrixaredenotedbylowercaseletters.
(E.g)matrix[A]withelementsa
ij










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mnijmm
nij
inij
aaaa
aaaa
aaaa
21
22221
1211
...
...

i denotes row number which goes from 1 to m
j denotes column number goes from 1 to n
A
mxn=

7. TYPES OF MATRICES
1.ColumnMatrixorVectorMatrix:
“AmatrixhavingonlyonecolumniscalledaColumn
matrix.Thenumberofrowsmaybeanyintegerbutthe
numberofcolumnsisalways1.”(i.e.)itismatrixofmx1.
(E.g.)
arecolumnmatrices.
(3x1)(2x1)









2
4
1 
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3
1 

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





1
21
11
ma
a
a


8. 2.RowMatrixorVectorMatrix:
“AmatrixhavingonlyonerowiscalledaRowmatrix.
Anynumberofcolumnsbutonlyonerow.”(i.e.)itisamatrix
of1xn.
(E.g.) (1x3) (1x4)
isaRowmatrices.
3.RectangularMatrix:
“Amatrixinwhichthenumberofrowsisnotequalto
thenumberofcolumnsiscalledarectangularmatrix”.
(E.g.)
(4x2) (2x5)
arerectangularmatrices. 611  2530  
n
aaaa
1131211
 

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67
77
73
11 
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
03302
00111 nm

9. 4.SquareMatrix:
“Amatrixinwhichthenumberofrowsequalto
numberofcolumnsiscalledaSquarematrix.”(asquare
matrixAhasanorderofm)
(E.g.) (2x2) (3x3)
isasquarematrices.
5.DiagonalMatrix:
“Asquarematrixinwhichalltheelementsexceptthe
diagonalelementsarezeroiscalledadiagonalmatrix”.
(E.g.)
(3x3) (4x4)
arediagonalmatrices.





03
11 
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166
099
111 
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100
020
001 
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

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
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9000
0500
0030
0003

10. 6.UnitorIdentityMatrix:
“Adiagonalmatrixinwhicheachdiagonalentryis
unity,iscalledUnitmatrixoridentitymatrix(I
n).”(i.e.)one
ondiagonalelements.
(E.g.) (2x2) (4x4)
isaidentitymatrices.
7.NullMatrix:
“Amatrixinwhicheachelementiszeroiscalledanull
orazeromatrixanddenotedbyO”.
(E.g.)
(1x3) (3x3)
arenullmatrices.











1000
0100
0010
0001 
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10
01 

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0
0
0 


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





000
000
000

11. 8.ScalarMatrix:
“Adiagonalmatrixinwhichallthediagonalelements
areequal,iscalledScalarmatrix.”Ascalarisdefinedasa
singlenumberorconstant.
(E.g.) (3x3) (4x4)
isascalarmatrices.
9.TriangularMatrix:
“Asquarematrixinwhicheachelementbeloworabove
themaindiagonalareallzeroiscalledaTriangularmatrix”.
(E.g.)









100
010
001 
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6000
0600
0060
0006 
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325
012
001 
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325
012
001 
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300
610
981

12. 9a.UpperTriangularMatrix:
“Asquarematrixinwhicheachelementbelowthe
maindiagonalareallzeroiscalledaUpperTriangular
matrix”.
(E.g.)
isauppertriangularmatrices.
9b.LowerTriangularMatrix:
“Asquarematrixinwhicheachelementabovethemain
diagonalareallzeroiscalledaLowerTriangularmatrix”.
(E.g.)


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


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325
012
001 
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300
810
781 
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3000
8700
4710
4471

13. Problems of Matrices -Addition

14. Problems of Matrices -Addition

15. Problems of Matrices -Subtraction

16. Problems of Matrices -Subtraction

17. Problems of Matrices –Addition & Subtraction
(Homework problems)

18. Problems of Matrices -
Multiplication

19. Problems of Matrices -
Multiplication

20. Problems of Matrices –Multiplication
(Homework problems)

21. Equality of Matrices

22. Equality of Matrices

23. TransposeofaMatrix:
“ThematrixobtainedfromAbyinterchangingtherows
intocolumnsiscalledthetransposeofamatrixAanditis
denotedbyA
I
(or)A
T
”.
(E.g.)IfA=2-13,thenA
I
=21
124 -12
34
LetAandBaretwomatricesofsameorder,then
i)(A+B)
I
=A
I
+B
I
ii)(A
I
)
I
=A

24. Transpose of Square Matrix

25. Transpose of Square Matrix

26. Transpose of Square Matrix

27. DeterminantsofaMatrix:
“ToeverysquarematrixAweassociateaunique
numbercalledthedeterminantofthematrixA.The
determinantofsquarematrixAisdenotedA.
(E.g.)IfAisansquarematrixwithordernxn,thenAiscalled
determinantofordern)or)nthorderdeterminant.
EvaluationofDeterminant:
LetA=a
1b
1
a
2b
2be2x2matrix,thenAwillbe
A=a
1b
1=a
1b
2–a
2b
1isvalueofA.
a
2b
2

28. Letusconsider3x3matrix,thendeterminantofAwillbe–
-multiplyeachelementof1
st
rowbycorresponding
determinantof2
nd
orderbydeletingtherowandcolumnof
thatelement&takeaproductbyaddingsignsalternatively(+,
-).
3x3matrixwithalternativelysignare +-+
-+-
+-+
Properties of Determinant:
1. The value of the determinant remains same if the rows and
columns are interchanged.
2. If two rows (or columns) of a determinant is interchanged,
the value of the determinant changes by a sign.

29. 3. If in a determinant any two rows (or columns) are identical,then
the value of the determinant is zero.
4. If in a determinant the elements of any row (or column) are
multiplied by the same scalar, say k, then the value of new
determinant is k times the given determinant.
5. If in a determinant the elements of any row (or column) is made
up of a sum of two quantities, then the determinant can be
expressed as the sum of two determinants of the same order.
6. If each element of a row (or column) is multiplied by a scalar
and added to the corresponding elements of any other row (or
column), then the value of determinant is unaltered.
7. If in a determinant all the elements of any row (or column) are
zerosthen the value of the determinant is zero.
8. If all the elements on one side of the principal diagonal of a
determinant are zeros, then the value of the determinant is the
product of the elements of the principal diagonal.

30. Determinants of Matrix

31. Determinants of Matrix

32. Determinants of Matrix

33. Minors, Cofactors & Adjoint of
Matrix

34. Minors, Cofactors & Ad joint of
Matrix

35. Ad joint of Matrix

36. Inverse of a Matrix

37. Inverse of a Matrix

38. Minor, Cofactor, Ad joint & Inverse
of a Matrix –Homework problems

39. Cramer’s Rule

40. Cramer’s Rule
ExerciseProblems:
1.SolvethefollowingequationsbyCramer’sRule.
2x–3y+1=0,3x+y–1=0
2.SolvethefollowingequationsbyCramer’sRule.
5x –y –4z = 5, 2x + 3y + 5z = 2 & 7x –2y + 6z = 5
3.SolvethefollowingequationsbyCramer’sRule.
4x + y = 7, 3y + 4z = 5 & 3z + 5x = 2

41. Cramer’s Rule
HomeworkProblems:
1.SolvethefollowingequationsbyCramer’sRule.
5x –y + z = 4, 3x + 2y -5z = 2 & x + 3y –2z = 5
2.SolvethefollowingequationsbyCramer’sRule.
x –y + 2z = 3, 2x + z = 1 & 3x + 2y + z = 4

42. Application Problems
ExerciseProblems:
1.TherearetwofamiliesAandB.Therearetwomen,
threewomenandonechildinFamilyAandhusband,
wifeand2childreninFamilyB.Thedailyintakeof
caloriesasrecommendedbyW.H.O.ismen–2400,
women–1900andchildren–1800.Forproteinintake
men–55gms,women–45gmsandchildren–33
gms.Usingmatrices,calculatethedailytotal
requirementsofcalories&proteinforeachofthetwo
families.

43. Application Problems
ExerciseProblems:
2.Asalespersonhasthefollowingrecordofsalesfor
themonthofJanuary,February,March1996forthree
productsA,BandC.heispaidacommissionatfixed
rateperunitbutatvaryingratesforproductsA,Band
C.
FindtherateofcommissionpayableonA,BandC
perunitsold.
Month Sales in Units Commission
(Rs.)
A B C
January 9 10 2 800
February15 5 4 900
March 6 10 3 850

44. Application Problems
HomeworkProblems:
1.TwofirmsXandY,eachmanufacturetwo
instrumentsPandQ.Thedailyproductionof
instrumentsisasfollows.FirmXproduces2unitsofP
and4unitsofQ.FirmYproduces1unitofPand3
unitsofQ.FirmXworkfor5daysinaweekandY
worksfor6daysinthework.Calculatetheweekly
productionvolume.

45. End of the Unit 4
Thank You