Classification Matrices execute measurements, for example, Log-Loss, Accuracy, AUC(Area under Curve), and so on. Another case of metric for assessment of AI calculations is exactness, review, which can be utilized for arranging calculations principally utilized via web indexes.

1. Matrix Introduction
Data science and AI Certification Course
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2. Matrices - Introduction
Matrix algebra has at least two advantages:
•Reduces complicated systems of equations to simple
expressions
•Adaptable to systematic method of mathematical treatment
and well suited to computers
Definition:
A matrix is a set or group of numbers arranged in a squareor
rectangular array enclosed by two brackets
[1 -1]
ë
ê-3 û ë û0ú êc dú
é 4 2ù éa bù
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3. Matrices - Introduction
Properties:
•A specified number of rows and a specified numberof
columns
•Two numbers (rows x columns) describe the dimensions
or size of the matrix.
Examples:
3x3 matrix
2x4 matrix
1x2 matrix
ú
úû
ê
êë
5úê4
é1 2 4ù
-1
3 3 3 ûë 2 úê0
é1 1 3 -3ù
0 3
[1 -1]
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4. Matrices - Introduction
A matrix is denoted by a bold capital letter and theelements
within the matrix are denoted by lower case letters
e.g. matrix [A] with elements aij
ú
úê
êë mn úûa aa
êa m2m1
éa11
êa21
! ú
aij ain ù
aij a2n ú
!
ij
a12...
a22...
!ê !
i goes from 1 to m
j goes from 1 to n
mxnA =
mAn
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5. Matrices - Introduction
TYPES OF MATRICES
1. Column matrix or vector:
The number of rows may be any integer but the number of
columns is always 1
ê ú
êë2úû
ê4ú
ë û
ê-3ú
é 1 ù
êë
ê ú
am1úû
êa21
é1ù éa11
ù
ú
ê ú
!
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6. Matrices - Introduction
TYPES OF MATRICES
2. Row matrix or vector
Any number of columns but only one row
[0 3 5 2]
a1n ]
[1 1 6]
a12[a11
a13 !
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7. Matrices - Introduction
TYPES OF MATRICES
3. Rectangular matrix
Contains more than one element and number of rows is not
equal to the number of columns
ûë 6 úê7
ú
-7ú
ê
ê7
ê3 7 ú
é1 1 ù
ûë 0úê2
é1 1 1 0 0ù
0 3 3
m ¹ n Visit: Learnbay.co

8. Matrices - Introduction
TYPES OF MATRICES
4. Square matrix
The number of rows is equal to the number of columns
A has an order of m)
ë3 0û
(a square matrix
é1 1ù
ê ú ú
1úû
ê
êë6
0úê9
é1 1 1ù
9
6
m x m
The principal or main diagonal of a square matrix is composed of all
elements aij for which i=j
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9. Matrices - Introduction
TYPES OF MATRICES
5. Diagonal matrix
A square matrix where all the elements are zero except thoseon
the main diagonal
ú
1úû
ê
êë0
0úê0
é1 0 0ù
2
0 ûë 9úê0
ú
0ú
ê
ê0
0úê0
é3 0 0 0ù
3 0
0 5
0 0
i.e. aij =0 for all i =j
aij = 0 for some or all i =j
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10. Matrices - Introduction
TYPES OF MATRICES
6. Unit or Identity matrix - I
A diagonal matrix with ones on the maindiagonal
ë û1úê0
ú
0ú
ê
ê0
0úê0
é1 0 0 0ù
1 0
0 1
0 0
ë0 1û
é1 0ù
ê ú
i.e. aij =0 for all i =j
aij = 1 for some or all i =j
ê
ij
aij ûë 0
éa 0 ù
ú
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11. Matrices - Introduction
TYPES OF MATRICES
7. Null (zero) matrix - 0
All elements in the matrix are zero
ê ú
êë0úû
ê0ú
é0ù
ú
0úû
ê
êë0
0úê0
é0 0 0ù
0
0
aij = 0 For all i,j
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12. Matrices - Introduction
TYPES OF MATRICES
8. Triangular matrix
A square matrix whose elements above or below the main
diagonal are all zero
ú
3úû
ê
êë5
0úê2
é1 0 0ù
1
2
ê
êë5
ê2 ú
3úû
6ú0ú ê0
é1 0 0ù é1 8 9ù
1 ú ê 1
2 3úû êë0 0
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13. Matrices - Introduction
TYPES OF MATRICES
8a. Upper triangular matrix
A square matrix whose elements below the main
diagonal are all zero
i.e. aij = 0 for all i >j
ù
ú
3úû
ê
êë0
8úê0
1 8 7
1
0
ûë 3úê0
ú
8ú
ê
ê0
4úê0
é1 7 4 4ù
1 7
0 7
0 0
ê
ë ij û
ij ij ij
ú
é
a ú
aéa a ù
aij aij ú
0ê 0
ê 0
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14. Matrices - Introduction
TYPES OF MATRICES
8b. Lower triangular matrix
A square matrix whose elements above the main diagonal are all
zero
i.e. aij = 0 for all i <j
ú
3úû
ê
êë5
0úê2
é1 0 0ù
1
2
ê ijij ú
êëaij aij aij
úû
a
êa
éaij
0 ú
0 ù0
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15. Matrices – Introduction
TYPES OF MATRICES
9. Scalar matrix
A diagonal matrix whose main diagonal elements are
equal to the same scalar
A scalar is defined as a single number or constant
ú
1úû
ê
êë0
0úê0
0 0ù
1
0
ûë 6úê0
0ú
0ú
ê0
ê0
é6 0 0 0ù
ê 6 0 ú
0 6
0 0i.e. a = 0 for all i = jij
aij = a for all i =j
ij ú
aij
úû
a
éaij
0
ê
êë0
0 úê 0
0 ù0 é1
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16. Matrices
´Matrix Operations
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17. Matrices - Operations
EQUALITY OFMATRICES
Two matrices are said to be equal only when all
corresponding elements are equal
Therefore their size or dimensions are equal as well
ú
3úû
ê
êë5
ê2 ú
3úû
ê
êë5
0ú0ú ê2
é1 0 0ù é1 0 0ù
1 1
2 2
A = B = A = B
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18. Matrices - Operations
Some properties of equality:
•If A = B, then B = A for all A and B
•If A = B, and B = C, then A = C for all A, B andC
ú
3úû
ê
êë5
0úê2
é1 0 0ù
1
2
A = B =
ê 21 23ú
b33úûêëb31
b úb b
éb11 b12 b13 ù
ê
22
b32
If A = B then aij = bij Visit: Learnbay.co

19. Matrices - Operations
ADDITION AND SUBTRACTION OF MATRICES
The sum or difference of two matrices, A and B of the same
size yields a matrix C of the same size
+ bijcij = aij
Matrices of different sizes cannot be added or subtracted
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20. Matrices - Operations
Commutative Law:
A + B = B + A
Associative Law:
A + (B + C) = (A + B) + C = A + B +C
ûû ëû ëë 9úê- 23ú
5 6ù é 8 8 5ù
ê- 4 - 2
=
-76 úê2
é7 3 -1ù
+
é 1
-5
A
2x3
B
2x3
C
2x3 Visit: Learnbay.co

21. Matrices - Operations
A + 0 = 0 + A = A
A + (-A) = 0 (where –A is the matrix composed of –aij as elements)
ûû ëû ëë -1ú8ú ê27ú ê1ê3
é6 4 2ù
-
é1 2 0ù
=
é5 2 2 ù
2 0 2
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22. Matrices - Operations
SCALAR MULTIPLICATION OF MATRICES
Matrices can be multiplied by a scalar (constant or single
element)
Let k be a scalar quantity; then
kA =Ak
Ex. If k=4 and
ë û1 úê4
ú
-3úê2
é3 -1ù
ê2 1 ú
A=ê
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23. Matrices - Operations
úê
ú=ê
ë û ë û
é12 -4ù
ú´4=ê ú
-3ú ê8 -12ú
ë16 4 û
ê2
1 ú ê8 4 ú
é3 -1ù
ê4 1 ú ê4 1 ú
-3úê2
é3 -1ù
ê2 1 ú ê2
4´ê
Properties:
• k (A + B) = kA + kB
• (k + g)A = kA + gA
• k(AB) = (kA)B = A(k)B
• k(gA) = (kg)A
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24. Matrices - Operations
MULTIPLICATION OF MATRICES
The product of two matrices is another matrix
Two matrices A and B must be conformable for multiplication to
be possible
i.e. the number of columns of A must equal the number of rows
of B
Example.
A x B = C
(1x3) (3x1) (1x1)
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25. Matrices - Operations
B x A = Not possible!
(2x1) (4x2)
= Not possible!A x B
(6x2) (6x3)
Example
A x B
(2x3) (3x2)
= C
(2x2)
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26. Matrices - Operations
ú
ùú =úê
ë 21 22 û
1211
ë 32 û31
21
ë 21 22 23 û
131211
cêc
éc c
ê b úb
bêb
ù
éb11 b12 ù
22 úa aêa
éa a a
(a11 ´ b11) + (a12 ´ b21) + (a13 ´ b31) = c11
(a11 ´ b12) + (a12 ´ b22) + (a13 ´ b32) = c12
(a21 ´ b11) + (a22 ´ b21) + (a23 ´ b31) = c21
(a21 ´b12) + (a22 ´b22) + (a23 ´b32) = c22
Successive multiplication of row i of A with column j of
B – row by column multiplication
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27. Matrices - Operations
úú ë(4´4) + (2´6) + (7´5)
(1´8) + (2´2) + (3´3)ù
(4´8) + (2´ 2) + (7´3)û
é(1´4) + (2´6) + (3´5)
êë5 3úû
2ú = êú
ê6
7ûêë4
3ù
é4 8ù
é1 2
ê 2
=
é31 21ù
ê63 57ú
ë û
ê0
Remember also:
IA = A
é1
ûë û ë1ú ê63 57ú
0ù é31
ûë
ê63 57ú
21ù
=
é31 21ù
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28. Matrices - Operations
Assuming that matrices A, B and C are conformable for
the operations indicated, the following are true:
1. AI = IA = A
2. A(BC) = (AB)C = ABC - (associative law)
3. A(B+C) = AB + AC - (first distributive law)
4. (A+B)C = AC + BC - (second distributive law)
Caution!
1. AB not generally equal to BA, BA may not be conformable
2. If AB = 0, neither A nor B necessarily = 0
3. If AB = AC, B not necessarily = C
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29. Matrices - Operations
AB not generally equal to BA, BA may not be conformable
ûë ûë û ë 0úê10
û
2ù
=
é23 6ù
ê0 2úê5 0ú
ë ûë û ë
4ùé1
ê15 20úê5 0úê0 2ú
ë û
2ùé3 4ù
=
é 3 8 ù
ê0 2ú
ë û
4ù
ê5 0ú
2ù
ST =
é3
TS =
é1
S =
é3
T =
é1
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30. Matrices - Operations
û ë ûë ûë0úê- 2 -3úê0 ê0 0ú
If AB = 0, neither A nor B necessarily = 0
é1 1ùé 2 3 ù
=
é0 0ù
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31. Matrices - Operations
TRANSPOSE OF AMATRIX
If :
ë û
é
1úê5
2 4 7ù
3
3
2
A= A =
2x3
ê ú
êë7 1úû
3ú3
2
TT
A = ê4A =
Then transpose of A, denoted ATis:
é2 5ù
jiija = aT
For all i and j
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32. Matrices - Operations
To transpose:
Interchange rows and columns
The dimensions of AT are the reverse of the dimensions ofA
ûë
é
1úê5
2 4 7ù
3
3
2A= A =
ê ú
êë7 1úû
2
3
é2 5ù
T ê 3úT
A = 4A =
2 x 3
3 x 2
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33. Matrices - Operations
Properties of transposed matrices:
1. (A+B)T = AT + BT
2. (AB)T = BT AT
3. (kA)T = kAT
4. (AT)T = A
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34. Matrices - Operations
1. (A+B)T = AT + BT
ûû ëû ëë 9ú3ú = ê-2
5 6ù é 8 8 5ù
- 2 -76 ú + ê- 4ê2
é7 3 -1ù é 1
-5 ê ú
êë5 9 úû
-7úê8
é8 -2ù
ú ê ú
3 úû êë5 9 úû
- 2ú = ê8 -7ú
ú ê
6 úû êë6
ê
êë-1
-5ú + ê5ê 3
2 ù é1 - 4ù é8 -2ùé 7
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35. Matrices - Operations
(AB)T = BTAT
2ú = [2 8][1 1 2]ê1ê ú
êë0 3úû
é1 0ù
ë8û
é2ù
êë2úû
ú
ê1ú = ê ú Þ [2 8]3ûê ú
0ù
é1ù
ë0
é1 1
ê 2
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36. Matrices - Operations
SYMMETRIC MATRICES
A Square matrix is symmetric if it is equal to its
transpose:
A = AT
ë û
êb dú
ë û
=
éa bù
êb dú
bù
A =
éa
AT
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37. Matrices - Operations
When the original matrix is square, transposition does not
affect the elements of the main diagonal
ë û
êb dú
ë û
=
éa cù
êc dú
bù
A =
éa
AT
The identity matrix, I, a diagonal matrix D, and a scalar matrix, K,
are equal to their transpose since the diagonal is unaffected.
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38. Matrices - Operations
INVERSE OF AMATRIX
Consider a scalar k. The inverse is the reciprocal or division of 1
by the scalar.
Example:
k=7 the inverse of k or k-1 = 1/k =1/7
Division of matrices is not defined since there may be AB = AC
while B = C
Instead matrix inversion is used.
The inverse of a square matrix, A, if it exists, is the unique matrix
A-1 where:
AA-1 = A-1 A =I
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39. Matrices - Operations
Example:
û
é
3 úê- 2
ë û
-1ù
ê2 1ú
3 1ù2
2
A-1
=
é 1
A= A =
û ë ûë ûë 3 ú ê0 1ú
-1ù
=
é1
1úê- 2ê2
ë ûë û ë û
é3 1ùé 1 0ù
1ú = ê0 1ú3 úê2ê- 2
ë
é 1 -1ùé3 1ù é1 0ù
Because:
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40. Matrices - Operations
Properties of the inverse:
( AB)-1
= B-1
A-1
( A-1
)-1
= A
( AT
)-1
= (A-1
)T
(kA)-1
=
1
A-1
k
A square matrix that has an inverse is called a nonsingularmatrix
A matrix that does not have an inverse is called a singular matrix
Square matrices have inverses except when the determinant is zero
When the determinant of a matrix is zero the matrix is singular Visit: Learnbay.co

41. Matrices - Operations
DETERMINANT OF AMATRIX
To compute the inverse of a matrix, the determinant is required
Each square matrix A has a unit scalar value called the
determinant of A, denoted by det A or |A|
2
6 5
2ù
ê6 5ú
ë û
A =
1
A =
é1
If
then
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42. Matrices - Operations
If A = [A] is a single element (1x1), then the determinant is
defined as the value of the element
Then |A| =det A = a11
If A is (n x n), its determinant may be defined in terms of order
(n-1) or less.
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43. Matrices - Operations
MINORS
If A is an n x n matrix and one row and one column are deleted,
the resulting matrix is an (n-1) x (n-1) submatrix of A.
The determinant of such a submatrix is called a minor of A and
is designated by mij , where i and j correspond to thedeleted
row and column, respectively.
mij is the minor of the element aij inA.
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44. Matrices - Operations
ú
ê 21 22 23 ú
éa11 a12 a13 ù
A = êa a a
êëa31 a32 a33 úû
Each element in A has a minor
Delete first row and column from A .
The determinant of the remaining 2 x 2 submatrix is the minor
of a11
eg.
32
11
a a
m =
a22 a23
33 Visit: Learnbay.co

45. Matrices - Operations
Therefore the minor of a12 is:
And the minor for a13 is:
31
12
a a
m =
a21 a23
33
31
13
a a
m =
a21 a22
32
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46. Matrices - Operations
COFACTORS
The cofactor Cij of an element aij is definedas:
C = (-1)i+ j
m
ij ij
When the sum of a row number i and column j is even, cij = mij and
when i+j is odd, cij =-mij
c (i = 1, j =1) = (-1)1+1
m = +m
11 11 11
c (i = 1, j = 2) = (-1)1+2
m = -m
12 12 12
c (i = 1, j = 3) =(-1)1+3
m = +m
13 13 13
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47. Matrices - Operations
DETERMINANTS CONTINUED
The determinant of an n x n matrix A can now be defined as
A = det A = a11c11 + a12c12 +"+ a1nc1n
The determinant of A is therefore the sum of the products of the
elements of the first row of A and their corresponding cofactors.
(It is possible to define |A| in terms of any other row or column
but for simplicity, the first row only is used)
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48. Matrices - Operations
Therefore the 2 x 2 matrix :
ú
ë 21 22 ûaêa
a12 ù
A =
éa11
= a22
Has cofactors :
c11 = m11 = a22
And:
2121c12 = -m12 =- a = -a
And the determinant of A is:
-a12 a21A = a11c11 + a12c12 = a11a22
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49. Matrices - Operations
Example 1:
ë û2úê1
1ù
A =
é3
A = (3)(2) -(1)(1) = 5
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50. Matrices - Operations
For a 3 x 3 matrix:
ú
ê 21 22 23 ú
éa11 a12 a13 ù
A = êa a a
êëa31 a32 a33 úû
The cofactors of the first row are:
22 3121 32
31
13
31
12
23 3222 33
32
11
a a
c
a a
c
a a
c
= a a - a a=
a21 a22
32
= -(a21a33 - a23a31)= -
a21 a23
33
= a a - a a=
a22 a23
33
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51. Matrices - Operations
The determinant of a matrix Ais:
A = a11c11 + a12c12 = a11a22 - a12a21
Which by substituting for the cofactors in this case is:
-a22a31)- a23a32 ) - a12 (a21a33 - a23a31) + a13(a21a32A =a11 (a22 a33
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52. Matrices - Operations
Example 2:
ú
1úû
3ú
é 1 0 1ù
A = ê 0 2
ê
êë-1 0
A = (1)(2- 0) - (0)(0+ 3)+ (1)(0+ 2) = 4
-a22a31)- a23a31) + a13(a21a32- a23a32 ) - a12 (a21a33A =a11 (a22 a33
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