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1. 1
Lecture 1
Matrices and Determinants
By KisorMukhopadhyay, PrabhuJagatbandhuCollege,
Module I

2. 2
1.1Matrices
1.2Operations of matrices
1.3Types of matrices
1.4Properties of matrices
1.5Determinants
1.6Inverse of a 33 matrix

3. 3237
115
A




1.1 Matrices131
214
476
B






Both Aand Bare examples of matrix. A matrix
is a rectangular array of numbers enclosed by a
pair of bracket.
Why matrix?

4. 4
How about solving7,
35.



 

xy
xy 27,
242,
54101,
365.









 
xyz
xyz
xyz
xyz
Consider the following set of equations:
It is easy to show that x = 3and
y = 4.
Matrices can help…
1.1 Matrices

5. 511 12 1
21 22 2
12







n
n
m m mn
aaa
aaa
A
aa a
In the matrix
▪numbers a
ij are called elements. First subscript
indicates the row; second subscript indicates
the column. The matrix consists of mnelements
▪It is called “the m nmatrix A = [a
ij]” or simply
“the matrix A ” if number of rows and columns
are understood.
1.1 Matrices

6. 6
Square matrices
▪When m =n, i.e.,
▪A is called a “square matrix of order n” or
“n-square matrix”
▪elements a
11, a
22, a
33,…, a
nncalled diagonal
elements.
▪ is called the traceof A.11 12 1
21 22 2
12







n
n
n n nn
aaa
aaa
A
aaa ...
11 22
1
 


n
ii nn
i
aaaa
1.1 Matrices

7. 7
Equal matrices
▪Two matrices A = [a
ij] and B = [b
ij] are said to
be equal (A = B) iff each element of A is equal
to the corresponding element of B, i.e., a
ij = b
ij
for 1 im, 1 jn.
▪iffpronouns “if and only if”
if A = B, it implies a
ij = b
ij for 1 im, 1 jn;
if a
ij = b
ij for 1 im, 1 jn, it implies A = B.
1.1 Matrices

8. 8
Equal matrices
Given that A = B, find a, b, cand d.
1.1 Matrices10
42
A



 ab
B
cd




Example: and
if A = B, then a= 1, b= 0, c= -4and d= 2.

9. 9
Zero matrices
▪Every element of a matrix is zero, it is called
a zero matrix, i.e., 000
000
000






A
1.1 Matrices

10. 10
Sums of matrices
1.2 Operations of matrices
▪If A = [a
ij
]and B = [b
ij
]are m nmatrices,
then A + Bis defined as a matrix C = A + B,
where C= [c
ij
], c
ij
= a
ij
+ b
ij
for 1 im, 1 jn.123
014




A 230
125




B
Example: if and
Evaluate A + Band A –B.122330353
0(1)1245139
  
 
  
  
AB 122330113
0(1)1245111
  
 
  
   
AB

11. 11
Sums of matrices
1.2 Operations of matrices
▪Two matrices of the sameorder are said to
be conformablefor addition or subtraction.
▪Two matrices of differentorders cannot be
added or subtracted, e.g.,
are NOT conformable for addition or
subtraction.237
115


 131
214
476






12. 12
Scalar multiplication
1.2 Operations of matrices
▪Let lbe any scalar andA = [a
ij
]is anm n
matrix. ThenlA = [la
ij
] for 1 im, 1 jn,
i.e., each element inAis multiplied byl.123
014




A
Example: . Evaluate 3A.313233369
3
3031340312
  

  
  
A
▪In particular, l  1, i.e., A = [a
ij
]. It’s called
thenegativeof A. Note: A A = 0is a zero matrix

13. 13
Properties
1.2 Operations of matrices
Matrices A, Band C are conformable,
▪A + B = B + A
▪A + (B +C)= (A+ B)+C
▪l(A+ B) = lA+ lB, where lis a scalar
(commutative law)
(associative law)
Can you prove them?
(distributive law)

14. 14
Let C= A + B, so c
ij
= a
ij
+b
ij
.
Consider lc
ij
= l(a
ij
+b
ij
) = la
ij
+lb
ij
, we have,
lC= lA+ lB.
Since lC = l(A+ B), so l(A+ B) = lA+ lB
Example: Prove l(A+ B) = lA+ lB.
Properties
1.2 Operations of matrices

15. 15
Matrix multiplication
1.2 Operations of matrices
▪If A = [a
ij
]is a m pmatrix and B = [b
ij
]is a
p nmatrix, then ABis defined as a m n
matrix C = AB, where C= [c
ij
] with11 22
1
...

 
p
ij ikkj ij i j ippj
k
cabababab 1 2 3
0 1 4




A 12
23
50






B
Example: , and C = AB.
Evaluate c
21
.12
123
23
014
50

 
 


 21
0(1)124522c
for 1 im, 1 jn.

16. 16
Matrix multiplication
1.2 Operations of matrices1 2 3
0 1 4




A 12
23
50






B
Example: , , Evaluate C = AB.11
12
21
22
1(1)223518
12
1223308123
23
0(1)124522014
50
0213403
c
c
c
c

 
 
 
 


 
 12
123 188
23
014 223
50
CAB

 
 
 
 



17. 17
Matrix multiplication
1.2 Operations of matrices
▪In particular, A is a 1mmatrix and
B is a m 1matrix, i.e., 11 1111 1221 1 1
1
...

 
m
kk mm
k
Cabababab  
11 12 1
...
m
Aaaa 11
21
1






m
b
b
B
b
then C = ABis a scalar.

18. 18
Matrix multiplication
1.2 Operations of matrices
▪BUT BA is a m mmatrix!  
11 1111 1112 111
21 2111 2112 211
11 12 1
1 111 112 11
...
  
  
  

  
  
  
m
m
m
m m m mm
b bababa
b bababa
BAaaa
b bababa
▪So AB BA in general !

19. 19
Properties
1.2 Operations of matrices
Matrices A, Band C are conformable,
▪A(B + C)= AB + AC
▪(A + B)C = AC+ BC
▪A(BC)= (AB) C
▪AB BA in general
▪AB = 0 NOT necessarily imply A = 0 orB = 0
▪AB = ACNOT necessarily imply B = C

20. 20
Properties
Let X = B + C, so x
ij
= b
ij
+c
ij
. Let Y = AX, then11
1 1 1
()
( )
nn
ij ikkj ikkj kj
kk
n n n
ikkj ikkj ikkj ikkj
k k k
yaxabc
abacabac

  
  
  


Example: Prove A(B + C)= AB + ACwhere A, B
andC are n-square matrices
So Y = AB + AC; therefore, A(B + C)= AB + AC
1.2 Operations of matrices

21. 21
1.3 Types of matrices
▪Identity matrix
▪The inverse of a matrix
▪The transpose of a matrix
▪Symmetric matrix
▪Orthogonal matrix

22. 22
▪A square matrix whose elements a
ij= 0,for
i >jis called upper triangular, i.e., 11 12 1
22 2
0
00






n
n
nn
aaa
aa
a
▪A square matrix whose elements a
ij= 0,for
i <jis called lower triangular, i.e., 11
21 22
12
00
0





n n nn
a
aa
aaa
Identity matrix
1.3 Types of matrices

23. 23
▪Both upper and lower triangular, i.e., a
ij= 0,for
i j, i.e., 11
22
00
00
00






 nn
a
a
D
a 1122
diag[,,...,]
nn
Daaa
Identity matrix
1.3 Types of matrices
is called a diagonal matrix, simply

24. 24
▪In particular, a
11= a
22 = … = a
nn = 1, the
matrix is called identity matrix.
▪Properties: AI = IA = A
Examples of identity matrices: and 10
01


 100
010
001





Identity matrix
1.3 Types of matrices

25. 25
▪AB BAin general. However, if two square
matrices Aand Bsuch that AB = BA, then A
and Bare said to be commute.
Can you suggest two matrices that must
commute with a square matrix A?
▪If Aand Bsuch that AB = -BA, then Aand B
are said to be anti-commute.
Special square matrix
1.3 Types of matrices
Ans: Aitself, the identity matrix, ..

26. 26
▪If matrices Aand Bsuch that AB = BA = I,
then Bis called the inverse of A(symbol:A
-1
);
and A is called the inverse of B(symbol:B
-1
).
The inverse of a matrix623
110
101
B






Show Bis the the inverse of matrix A.123
133
124
A






Example:100
010
001
ABBA






Ans: Note that
Can you show the
details?
1.3 Types of matrices

27. 27
The transpose of a matrix
▪The matrix obtained by interchanging the
rows and columns of a matrix Ais called the
transpose of A(write A
T
).
Example:
The transpose of Ais 123
456




A 14
25
36






T
A
▪For a matrix A = [a
ij], its transpose A
T
= [b
ij],
where b
ij= a
ji.
1.3 Types of matrices

28. 28
Symmetric matrix
▪A matrix Asuch that A
T
= A is called symmetric,
i.e., a
ji = a
ijfor all iand j.
▪A+ A
T
must be symmetric. Why?
Example: is symmetric.123
245
356






A
▪A matrix Asuch that A
T
= -A is called skew-
symmetric, i.e., a
ji = -a
ijfor all iand j.
▪A-A
T
must be skew-symmetric. Why?
1.3 Types of matrices

29. 29
Orthogonal matrix
▪A matrix Ais called orthogonal if AA
T
= A
T
A = I,
i.e., A
T
= A
-1
Example: prove that is
orthogonal.1/31/61/2
1/32/60
1/31/61/2
 




A
We’ll see that orthogonal matrix represents a
rotation in fact!
1.3 Types of matrices
Since, . Hence, AA
T
= A
T
A = I.1/31/31/3
1/62/61/6
1/201/2
T
A






Can you show the
details?

30. 30
▪(AB)
-1
= B
-1
A
-1
▪(A
T
)
T
= A and (lA)
T
= l A
T
▪(A + B)
T
= A
T
+ B
T
▪(AB)
T
= B
T
A
T
1.4 Properties of matrix

31. 31
1.4 Properties of matrix
Example: Prove (AB)
-1
= B
-1
A
-1
.
Since (AB)(B
-1
A
-1
)= A(B B
-1
)A
-1
= Iand
(B
-1
A
-1
)(AB)= B
-1
(A
-1
A)B = I.
Therefore, B
-1
A
-1
is the inverse of matrix AB.

32. 32
1.5 Determinants
Consider a 22matrix:11 12
21 22
aa
A
aa




Determinant of order 2
▪Determinant of A, denoted , is a number
and can be evaluated by 11 12
1122 1221
21 22
||
aa
A aaaa
aa
  ||A

33. 3311 12
1122 1221
21 22
||
aa
A aaaa
aa
 
Determinant of order 2
▪easy to remember (for order 2 only)..12
34
Example: Evaluate the determinant:12
14232
34

1.5 Determinants
+-

34. 34
1.5 Determinants
1.If every element of a row (column) is zero,
e.g., , then |A| = 0.
2.|A
T
| = |A|
3.|AB| = |A||B|
determinant of a matrix
=that of its transpose
The following properties are true for
determinants of anyorder.12
10200
00


35. 35
Example: Show that the determinant of any
orthogonal matrix is either +1or –1.
For any orthogonal matrix, A A
T
= I.
Since |AA
T
| = |A||A
T
| = 1and |A
T
| = |A|, so |A|
2
= 1or
|A| = 1.
1.5 Determinants

36. 36
1.5 Determinants
For any 2x2 matrix11 12
21 22
aa
A
aa




Its inverse can be written as 22 121
21 11
1aa
A
aaA






Example: Find the inverse of10
12
A




The determinant of Ais -2
Hence, the inverse of Ais1
10
1/21/2
A





How to find an inverse for a 3x3 matrix?

37. 37
1.5 Determinants of order 3
Consider an example:123
456
789
A






Its determinant can be obtained by: 123
451212
456369
787845
789
A  3366930
You are encouraged to find the determinant
by using other rows or columns

38. 38
1.6 Inverse of a 33 matrix
Cofactor matrix of 123
045
106
A






The cofactor for each element of matrix A:11
45
24
06
A 12
05
5
16
A 13
04
4
10
A 21
23
12
06
A 22
13
3
16
A 23
12
2
10
A 31
23
2
45
A 32
13
5
05
A 33
12
4
04
A

39. 39
Cofactor matrix of is then given
by:1 2 3
0 4 5
1 0 6
A





 2454
1232
254






1.6 Inverse of a 33 matrix

40. 40
1.6 Inverse of a 33 matrix
Inverse matrix of is given by:1 2 3
0 4 5
1 0 6
A





 1
2454 24122
11
1232 535
22
254 424
T
A
A

   
  
  
  
   
   1211611111
522322522
211111211





