The document discusses matrices and their operations. It defines what a matrix is, provides examples of different types of matrices, and covers key matrix operations like addition, subtraction, scalar multiplication, and matrix multiplication. It also defines important matrix concepts such as the transpose of a matrix, inverse of a matrix, and properties related to these operations and concepts.

1. MATHS MATHS
PRESENTATIONPRESENTATION
GROUP MEMBERS : DRISHTI (1838)
VANDANA (1831)
DIMPY(1833)

2.  Operations with Matrices
 Properties of Matrix Operations
 The Inverse of a Matrix
MATRICES

3.  Matrix:
A=[ aij ]=
[
a11 a12 a13 ⋯ a1n
a21 a22 a23 ⋯ a2n
a31 a32 a33 ⋯ a3n
⋮ ⋮ ⋮ ⋮
am1 am2 am3 ⋯ amn
]m×n
∈ M m×n
(i, j)-th entry: aij
row: m
column: n
size: m×n

4.  i-th row vector
ri=[ai1 ai2 ⋯ ain ]
 j-th column vector
c j=
[
c1j
c2j
⋮
cmj
]
row matrix
column matrix
 Square matrix: m = n

5.  Diagonal matrix:
A=diag (d1 ,d2 ,⋯ ,dn) =
[
d1 0 … 0
0 d2 ⋯ 0
⋮ ⋮ ⋱ ⋮
0 0 ⋯ dn
]∈M n×n
 Trace:
If A=[aij ]n×n
Then Tr ( A )=a11 +a22 +⋯+ann

6. Ex:
A=[1 2 3
4 5 6 ] =
[r1
r2
]
=[c1 c2 c3 ]
r1=[1 2 3 ], r2=[4 5 6 ]
c1=[1
4], c2=[2
5], c3=[3
6]
A=[1 2 3
4 5 6 ]
⇒
⇒

7. If A=[aij ]m×n , B=[bij ]m×n
 Equal matrix:
Then A=B if and only if aij =bij ∀ 1≤i≤m, 1≤ j≤n
 Ex 1: (Equal matrix)
A=[1 2
3 4 ] B=[a b
c d ]
If A=B
Then a=1, b=2, c=3, d=4

8.  Matrix addition:
If A=[aij ]m×n , B=[bij ]m×n
Then A+B=[ aij ]m×n +[ bij ]m×n=[ aij +bij ]m×n
 Ex 2: (Matrix addition)
[−1 2
0 1 ]+[ 1 3
−1 2 ]=[−1+1 2+3
0−1 1+2 ]=[0 5
−1 3 ]
[1
−3
−2]+
[−1
3
2 ]=
[1−1
−3+3
−2+2] =
[0
0
0]

9.  Matrix subtraction:
A−B=A+(−1)B
 Scalar multiplication:
If A=[aij ]m×n , c: scalar
 Ex 3: (Scalar multiplication and matrix subtraction)
A=
[1 2 4
−3 0 −1
2 1 2 ] B=
[2 0 0
1 −4 3
−1 3 2 ]
Find (a) 3A, (b) –B, (c) 3A – B
Then cA=[caij ]m×n

10. (a)
3A=3
[1 2 4
−3 0 −1
2 1 2 ]
(b)
−B=(−1)
[2 0 0
1 −4 3
−1 3 2 ]
(c)
3A−B=
[3 6 12
−9 0 −3
6 3 6 ]−
[2 0 0
1 −4 3
−1 3 2 ]
Sol:
=
[3 6 12
−9 0 −3
6 3 6 ]=
[
3(1) 3(2) 3(4)
3(−3) 3(0) 3(−1)
3(2) 3(1) 3(2) ]
=
[−2 0 0
−1 4 −3
1 −3 −2 ]
=
[ 1 6 12
−10 4 −6
7 0 4 ]

11.  Matrix multiplication:
If A=[aij ]m×n , B=[bij ]n× p
Then AB=[ aij ]m×n [bij ]n× p=[cij ]m×p
cij=∑
k=1
n
aik bkj =ai1 b1j +ai2 b2j+⋯+ain bnj
where
 Notes: (1) A+B = B+A, (2)AB≠BA
Size of AB
[
a11 a12 ⋯ a1n
⋮ ⋮ ⋮
ai1 ai2 ⋯ ain
⋮
an1
⋮
an2
⋯
⋮
ann
][
b11 ⋯ b1j ⋯ b1n
b21 ⋮ b2j ⋯ b2n
⋮ ⋮ ⋮ ⋮
bn1 ⋯ bnj ⋯ bnn
]=
[ci1 ci2 ⋯ cij ⋯ cin
]

12. A=
[−1 3
4 −2
5 0 ] B=[−3 2
−4 1 ]
 Ex 4: (Find AB)
Sol:
AB=
[
(−1)(−3)+(3)(−4) (−1)(2)+(3)(1)
(4)(−3)+(−2)(−4) (4)(2)+(−2)(1)
(5)(−3)+(0)(−4) (5)(2)+(0)(1) ]
=
[−9 1
−4 6
−15 10 ]

13.  Matrix form of a system of linear
equations:
{
a11 x1 +a12 x2+⋯+a1n xn =b1
a21 x1 +a22 x2+⋯+a2n xn =b2
⋮
am1 x1 +am2 x2+⋯+amn xn =bm
}=
=
=
A x b
m linear equations
Single matrix equation
Ax =b
m×nn×1 m×1[
a11 a12 ⋯ a1n
a21 a22 ⋯ a2n
⋮ ⋮ ⋮ ⋮
am1 am2 ⋯ amn
][
x1
x2
⋮
xn
]=
[
b1
b2
⋮
bm
]
⇓

14.  Partitioned matrices:
A=
[
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
]=
[A11 A12
A21 A22
]
submatrix
A=
[
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
]=
[
r1
r2
r3
]
A=
[
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
]=[c1 c2 c3 c4 ]

15.  Three basic matrix operators:
(1) matrix addition
(2) scalar multiplication
(3) matrix multiplication
 Zero matrix: 0m×n
 Identity matrix of order n: I n

16. Then (1) A+B = B + A
(2) A + ( B + C ) = ( A + B ) + C
(3) ( cd ) A = c ( dA )
(4) 1A = A
(5) c( A+B ) = cA + cB
(6) ( c+d ) A = cA + dA
If A,B,C ∈M m×n , c,d :scalar
 Properties of matrix addition and scalar multiplication:

17. If A∈Mm×n , c :scalar
Then (1) A+0m×n =A
(2) A+(− A)=0m×n
(3) cA=0m×n ⇒ c= 0 or A= 0m× n
 Notes:
(1) 0m×n: the additive identity for the set of all m×n matrices
(2) –A: the additive inverse of A
 Properties of zero matrices:

18. If A=
[
a11 a12 ⋯ a1n
a21 a22 ⋯ a2n
⋮ ⋮ ⋮ ⋮
am1 am2 ⋯ amn
]∈M m×n
Then AT
=
[
a11 a21 ⋯ am1
a12 a22 ⋯ am2
⋮ ⋮ ⋮ ⋮
a1n a2n ⋯ amn
]∈ M n×m
 Transpose of a matrix :
Transpose is the interchange of rows and columns of a
given matrix.

19. A=[2
8] (b) A=
[1 2 3
4 5 6
7 8 9 ] (c
)
A=
[0 1
2 4
1 −1]
Sol: (a)
A=[2
8] ⇒ A
T
=[2 8 ]
(b)
A=
[1 2 3
4 5 6
7 8 9 ] ⇒ AT
=
[1 4 7
2 5 8
3 6 9 ](c
)
A=
[0 1
2 4
1 −1] ⇒ AT
=[0 2 1
1 4 −1]
(a)
 Ex 8: (Find the transpose of the following matrix)

20. (1) ( A
T
)
T
=A
(2 ) ( A+B)
T
=A
T
+B
T
(3) (cA)T
=c ( AT
)
(4 ) ( AB )
T
=B
T
A
T
 Properties of transposes

21. A square matrix A is symmetric if A = AT
 Ex:
If A=
[1 2 3
a 4 5
b c 6 ] is symmetric, find a, b, c?
A square matrix A is skew-symmetric if AT
= –A
 Skew-symmetric matrix:
Sol:
A=
[1 2 3
a 4 5
b c 6 ] AT
=
[1 a b
2 4 c
3 5 6 ] a= 2,b= 3,c= 5
A=A
T
⇒
 Symmetric matrix:

22. If A=
[0 1 2
a 0 3
b c 0 ] is a skew-symmetric, find a, b, c?

Note:
AAT
is symmetric
Pf: (AAT
)T
=( AT
)T
AT
=AAT
AA∴
T
is symmetric
Sol:
A=
[0 1 2
a 0 3
b c 0 ] −AT
=
[0 −a −b
−1 0 −c
−2 −3 0 ]
A=−A
T
a=−1,b=−2,c=−3⇒
 Ex:

23. ab = ba (Commutative law for multiplication)
(1)If m≠ p, then AB is defined ,BA is undefined .
(3)If m=p=n,then AB∈M m×m ,BA∈ M m×m
(2)If m=p, m≠n, then AB∈M m×m ,BA∈ M n×n (Sizes are not the
same)
(Sizes are the same, but matrices are not equal)
 Real number:
 Matrix:
AB≠BA
m×nn× p
Three
situations:

24. A=[1 3
2 −1] B=[2 −1
0 2 ]
Sol:
AB=[1 3
2 −1 ][2 −1
0 2 ]=[2 5
4 −4 ]
 Note: AB≠BA
BA=[2 −1
0 2 ][1 3
2 −1]=[0 7
4 −2 ]
 Ex 4:
Sow that AB and BA are not equal for the matrices.
and

25. (Cancellation is not
valid)
ac=bc, c≠0
⇒ a=b (Cancellation law)
 Matrix:
AC=BC C ≠0
(1) If C is invertible, then A = B
 Real number:
A≠B(2) If C is not invertible, then

26. 26
/61
A=[1 3
0 1 ], B=[2 4
2 3 ], C=[ 1 −2
−1 2 ]
Sol:
AC=[1 3
0 1 ][ 1 −2
−1 2 ]=[−2 4
−1 2 ]
So AC=BC
But A≠B
BC=[2 4
2 3 ][ 1 −2
−1 2 ]=[−2 4
−1 2 ]
 Ex 5: (An example in which cancellation is not valid)
Show that AC=BC
Elementary Linear Algebra: Section 2.2, p.65

27. A∈M n×n
If there exists a matrix B∈ M n× n such that AB=BA=In ,

Note:
A matrix that does not have an inverse is called
noninvertible (or singular).
Consider
Then (1) A is invertible (or nonsingular)
(2) B is the inverse of A
 Inverse matrix:

28. If B and C are both inverses of the matrix A, then B = C.
Pf: AB=I
C (AB)=CI
(CA)B=C
IB=C
B=C
Consequently, the inverse of a matrix is unique.

Notes:(1) The inverse of A is denoted by A
−1
(2) AA
−1
=A
−1
A=I
 Thm 2.7: (The inverse of a matrix is unique)

29. [A ∣ I ]⃗Gauss-Jordan Elimination [I ∣ A−1
]
 Ex 2: (Find the inverse of the matrix)
A=[ 1 4
−1 −3 ]
Sol:
AX=I
[ 1 4
−1 −3 ][x11 x12
x21 x22
]=[1 0
0 1 ]
[ x11+4x21 x12+4x22
−x11−3x21 −x12−3x22
]=[1 0
0 1 ]
 Find the inverse of a matrix by Gauss-Jordan
Elimination:

30. (1)⇒[ 1 4 ⋮ 1
−1 −3 ⋮ 0 ]⃗
r12
(1)
, r21
(−4)
[1 0 ⋮ −3
0 1 ⋮ 1 ]
(2)⇒[1 4 ⋮ 0
−1 −3 ⋮ 1 ]⃗
r12
(1)
, r21
(−4)
[1 0 ⋮ −4
0 1 ⋮ 1 ]
⇒ x11=−3, x21=1
⇒ x12=−4, x22=1
X=A−1
=
[x11 x12
x21 x22
]=[−3 −4
1 1 ] ( AX=I=AA-1
)
Thus
⇒
x11 + 4x21 1
−x11 − 3x21 0
(1)
x12 + 4x22 0
−x12 − 3x22 1
(2)

31. ⃗r12
(1)
,r
21
(−4)
[1 0 ⋮ −3 −4
0 1 ⋮ 1 1 ]¿ A I I A−1
¿¿¿
If A can’t be row reduced to I, then A is singular.
 Note:

32. A=
[1 −1 0
1 0 −1
−6 2 3 ]
⃗R2−R1
[1 −1 0 ⋮ 1 0 0
0 1 −1 ⋮ −1 1 0
−6 2 3 ⋮ 0 0 1]
Sol:
[A ⋮ I ] =
[1 −1 0
1 0 −1
−6 2 3
⋮
⋮
⋮
1 0 0
0 1 0
0 0 1 ]
⃗R3+6R1
[1 −1 0
0 1 −1
0 −4 3
⋮
⋮
⋮
1 0 0
−1 1 0
6 0 1 ]
⃗−R3
[1 −1 0 ⋮ 1 0 0
0 1 −1 ⋮ −1 1 0
0 0 1 ⋮ −2 −4 −1 ]⃗R3+4R2
[1 −1 0 ⋮ 1 0 0
0 1 −1 ⋮ −1 1 0
0 0 −1 ⋮ 2 4 1 ]
 Ex 3: (Find the inverse of the following matrix)

33. ⃗R2+R3
[1 −1 0 ⋮ 1 0 0
0 1 0 ⋮ −3 −3 −1
0 0 1 ⋮ −2 −4 −1 ] ⃗R1+R2
[1 0 0 ⋮ −2 −3 −1
0 1 0 ⋮ −3 −3 −1
0 0 1 ⋮ −1 −4 −1 ]
So the matrix A is invertible, and its inverse is
A−1
=
[−2 −3 −1
−3 −3 −1
−2 −4 −1 ]
= [I ⋮ A−1
]
 Check:
AA−1
=A−1
A=I

34. (1) A
0
=I
(2) A
k
=AA⋯A⏟
k factors
(k> 0)
(3) Ar
⋅As
=Ar+s
r,s:integers
( A
r
)
s
=A
rs
(4) D=
[
d1 0 ⋯ 0
0 d2 ⋯ 0
⋮ ⋮ ⋱ ⋮
0 0 ⋯ dn
]⇒ Dk
=
[
d1
k
0 ⋯ 0
0 d2
k
⋯ 0
⋮ ⋮ ⋮
0 0 ⋯ dn
k ]
 Power of a square matrix:

35. If A is an invertible matrix, k is a positive integer, and c is a scalar
not equal to zero, then
(1) A
−1
is invertible and ( A
−1
)
−1
=A
(2) Ak
is invertible and ( Ak
)−1
=A−1
A−1
⋯A−1
⏟
k factors
=( A−1
)k
=A−k
(3) c A is invertible and (cA)−1
=
1
c
A−1
,c≠0
(4)A
T
is invertible and ( A
T
)
−1
=(A
−1
)
T
 Thm 2.8 ： (Properties of inverse matrices)

36.  Thm 2.9: (The inverse of a product)
If A and B are invertible matrices of size n, then AB is invertible and
(AB)
−1
=B
−1
A
−1
If AB is invertible, then its inverse is unique.
So (AB)−1
=B−1
A−1
Pf:
(AB)(B−1
A−1
)=A( BB−1
) A−1
=A(I )A−1
=( AI) A−1
=AA−1
=I
(B
−1
A
−1
)( AB)=B
−1
( A
−1
A)B=B
−1
(I )B=B
−1
( IB)=B
−1
B=I

Note:
(A1 A2 A3⋯An )
−1
=An
−1
⋯A3
−1
A2
−1
A1
−1

37.  Thm 2.10 (Cancellation properties)
If C is an invertible matrix, then the following properties hold:
(1) If AC=BC, then A=B (Right cancellation property)
(2) If CA=CB, then A=B (Left cancellation property)
Pf:
AC=BC
(AC )C−1
=(BC)C−1
A(CC−1
)=B(CC−1
)
AI=BI
A=B
(C is invertible, so C
-1
exists)

Note:If C is not invertible, then cancellation is not valid.

38.  Thm 2.11: (Systems of equations with unique solutions)
If A is an invertible matrix, then the system of linear equations
Ax = b has a unique solution given by
x=A
−1
b
Pf:
( A is nonsingular)
Ax=b
A−1
Ax=A−1
b
Ix=A−1
b
x=A−1
b
This solution is unique.
If x1 and x2 were two solutions of equation Ax=b .
then Ax1 =b=Ax2
⇒ x1=x2 (Left cancellation property)