The document discusses matrix multiplication and properties of matrices such as identity matrices, transposes, and powers of matrices. It also covers zero-one matrices, which have entries of only 0 and 1. Operations on zero-one matrices include join, meet, Boolean product, and Boolean powers. Boolean operations like AND and OR are used in place of multiplication and addition for zero-one matrix operations. Examples are provided to illustrate matrix multiplication and operations on zero-one matrices.

1. February 24, Applied Discrete Mathematics 1
Matrix MultiplicationMatrix Multiplication
Let us calculate the complete matrix C:Let us calculate the complete matrix C:










−
−−=
011
500
412
103
A








−=
43
10
12
B










=C
99
88
1515
-2-2
77
1515
2020
-2-2

2. February 24, Applied Discrete Mathematics 2
Identity MatricesIdentity Matrices
TheThe identity matrix of order nidentity matrix of order n is the nis the n××n matrixn matrix
IInn = [= [δδijij], where], where δδijij = 1 if i = j and= 1 if i = j and δδijij = 0 if i= 0 if i ≠≠ j:j:
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












=
1...00
...
...
...
0...10
0...01
A
Multiplying an mMultiplying an m××n matrix A by an identity matrix ofn matrix A by an identity matrix of
appropriate size does not change this matrix:appropriate size does not change this matrix:
AIAInn = I= ImmA = AA = A

3. February 24, Applied Discrete Mathematics 3
Powers and Transposes of MatricesPowers and Transposes of Matrices
TheThe power functionpower function can be defined forcan be defined for squaresquare
matrices. If A is an nmatrices. If A is an n××n matrix, we have:n matrix, we have:
AA00
= I= Inn,,
AArr
= AAA…A= AAA…A (r times the letter A)(r times the letter A)
TheThe transposetranspose of an mof an m××n matrix A = [an matrix A = [aijij], denoted by], denoted by
AAtt
, is the n, is the n××m matrix obtained by interchanging them matrix obtained by interchanging the
rows and columns of A.rows and columns of A.
In other words, if AIn other words, if Att
= [b= [bijij], then b], then bijij = a= ajiji forfor
i = 1, 2, …, n and j = 1, 2, …, m.i = 1, 2, …, n and j = 1, 2, …, m.

4. February 24, Applied Discrete Mathematics 4
Powers and Transposes of MatricesPowers and Transposes of Matrices
Example:Example:




−
=
411
302t
A








−=
43
10
12
A
A square matrix A is calledA square matrix A is called symmetricsymmetric if A = Aif A = Att
..
Thus A = [aThus A = [aijij] is symmetric if a] is symmetric if aijij = a= ajiji for allfor all
i = 1, 2, …, n and j = 1, 2, …, m.i = 1, 2, …, n and j = 1, 2, …, m.








−
−=
493
921
315
A








=
131
131
131
B
A is symmetric, B is not.A is symmetric, B is not.

5. February 24, Applied Discrete Mathematics 5
Zero-One MatricesZero-One Matrices
A matrix with entries that are either 0 or 1 is called aA matrix with entries that are either 0 or 1 is called a
zero-one matrixzero-one matrix. Zero-one matrices are often used. Zero-one matrices are often used
like a “table” to represent discrete structures.like a “table” to represent discrete structures.
We can define Boolean operations on the entries inWe can define Boolean operations on the entries in
zero-one matrices:zero-one matrices:
aa bb aa∧∧bb
00 00 00
00 11 00
11 00 00
11 11 11
aa bb aa∨∨bb
00 00 00
00 11 11
11 00 11
11 11 11

6. February 24, Applied Discrete Mathematics 6
Zero-One MatricesZero-One Matrices
Let A = [aLet A = [aijij] and B = [b] and B = [bijij] be m] be m××n zero-one matrices.n zero-one matrices.
Then theThen the joinjoin of A and B is the zero-one matrix withof A and B is the zero-one matrix with
(i, j)th entry a(i, j)th entry aijij ∨∨ bbijij. The join of A and B is denoted by. The join of A and B is denoted by
AA ∨∨ B.B.
TheThe meetmeet of A and B is the zero-one matrix with (i,of A and B is the zero-one matrix with (i,
j)th entry aj)th entry aijij ∧∧ bbijij. The meet of A and B is denoted by A. The meet of A and B is denoted by A
∧∧ B.B.

7. February 24, Applied Discrete Mathematics 7
Zero-One MatricesZero-One Matrices
Example:Example:








=
01
10
11
A








=
00
11
10
B
Join:Join:
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



=








∨∨
∨∨
∨∨
=∨
01
11
11
0001
1110
1101
BA
Meet:Meet:








=








∧∧
∧∧
∧∧
=∧
00
10
10
0001
1110
1101
BA

8. February 24, Applied Discrete Mathematics 8
Zero-One MatricesZero-One Matrices
Let A = [aLet A = [aijij] be an m] be an m××k zero-one matrix andk zero-one matrix and
B = [bB = [bijij] be a k] be a k××n zero-one matrix.n zero-one matrix.
Then theThen the Boolean productBoolean product of A and B, denoted byof A and B, denoted by
AAοοB, is the mB, is the m××n matrix with (i, j)th entry [cn matrix with (i, j)th entry [cijij], where], where
ccijij = (a= (ai1i1 ∧∧ bb1j1j)) ∨∨ (a(ai2i2 ∧∧ bb2i2i)) ∨∨ …… ∨∨ (a(aikik ∧∧ bbkjkj).).
Note that the actual Boolean product symbol has aNote that the actual Boolean product symbol has a
dot in its center.dot in its center.
Basically, Boolean multiplication works like theBasically, Boolean multiplication works like the
multiplication of matrices, but with computingmultiplication of matrices, but with computing ∧∧
instead of the product andinstead of the product and ∨∨ instead of the sum.instead of the sum.

9. February 24, Applied Discrete Mathematics 9
Zero-One MatricesZero-One Matrices
Example:Example:



=
11
01A



=
10
10B



=



∧∨∧∧∨∧
∧∨∧∧∨∧
=
10
10
)11()11()01()01(
)10()11()00()01(
BA

10. February 24, Applied Discrete Mathematics 10
Zero-One MatricesZero-One Matrices
Let A be a square zero-one matrix and r be a positiveLet A be a square zero-one matrix and r be a positive
integer.integer.
TheThe r-th Boolean powerr-th Boolean power of A is the Boolean productof A is the Boolean product
of r factors of A. The r-th Boolean power of A isof r factors of A. The r-th Boolean power of A is
denoted by Adenoted by A[r][r]
..
AA[0][0]
= I= Inn,,
AA[r][r]
= A= AοοAAοο……οοAA (r times the letter A)(r times the letter A)