My new article for 3D matrices

Open Access Library Journal 2017, Volume 4, e3593 ISSN Online: 2333-9721 ISSN Print: 2333-9705 3D Matrix Ring with a “Common” Multiplication Orgest Zaka Department of Mathematics, Faculty of Technical Science, University of Vlora “Ismail QEMALI”, Vlora, Albania How to cite this paper: Zaka, O. (2017) 3D Matrix Ring with a “Common” Multiplication. Open Access Library Journal, 4: e3593. https://doi.org/10.4236/oalib.1103593 Received: April 11, 2017 Accepted: May 12, 2017 Published: May 15, 2017 Copyright © 2017 by author and Open Access Library Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access Abstract In this article, starting from geometrical considerations, he was born with the idea of 3D matrices, which have developed in this article. A problem here was the definition of multiplication, which we have given in analogy with the usual 2D matrices. The goal here is 3D matrices to be a generalization of 2D matrices. Work initially we started with 3 × 3 × 3 matrix, and then we extended to m × n × p matrices. In this article, we give the meaning of 3D matrices. We also defined two actions in this set. As a result, in this article, we have reached to present 3-dimensional unitary ring matrices with elements from a field F. Subject Areas Algebra, Applied Statistical Mathematics, Geometry Keywords Linear Algebra, Matrices, Ring Theory 1. Introduction Based on the meaning of the addition and the multiplication of 2D matrices [1]-[6], this article stretches this sense, the idea, the addition and the multiplication of 3D matrices. Starting from geometrical considerations, concretely taking into account the cube, he was born with the idea of 3D matrices, which have developed in this article. A problem here was the definition of multiplication, for which we have acted pages, analogously acted as the columns, which we have given in analogy with the usual 2D matrices [6] [9]. The goal here is 3D matrices to be a generalization of 2D matrices. We proved that this set of two actions together in forming the “unitary ring” [7] [8] [10] [11]. In literature and in various mathematical forums, we noticed an interest in the 3D matrices, but on the other hand are missing results associated with them; this was a sufficient reason to explore. We introduced the meaning of the scalar multiplication, and finally DOI: 10.4236/oalib.1103593 May 15, 2017 O. Zaka we have shown that we have an F-module connected to this ring 3D matrices or vector spaces [8] [10]. As indications for this paper were simply geometric imaginations. Everything presented in this article are my results. 2. Addition of 3 × 3 × 3, 3-D Matrices over Field F, and the Addition Abelian Group of Their 3-D Matrices Imagining a parallelepiped, with born idea of 3D matrices, which are define as follows Definition 2.1 3-dimensional 3 × 3 × 3 matrice will call, a matrix which has: three horizontal layers (analogous to three rows), three vertical page (analogue with three columns in the usual matrices) and three vertical layers two of which are hidden. { } 3×3×3 ( F ) = ( aijk ) | aijk ∈ F and i = 1, 2, 3; j = 1, 2, 3; k = 1, 2, 3 The set of these matrices the write how: Definition 2.2 The addition of two matrices A3×3×3 , B3×3×3 ∈ 3×3×3 ( F ) we The appearance of these matrices will be as in Figure 1. { } C3×3×3 = ( cijk ) | cijk = aijk + bijk , ∀i, j , k ∈ {1, 2, 3} will call the matrix: The appearance of the addition of 3 × 3 × 3, 3D matrices, will be as in Figure 2, {( a ) | a = {( b ) | b } ∈ F for i = 1, 2, 3; j = 1, 2, 3; k = 1, 2, 3} where matrices A and B have the following appearance, A3×3×3 = ijk ijk B3×3×3 ijk ijk ∈ F for i = 1, 2, 3; j = 1, 2, 3; k = 1, 2, 3 113 123 122 112 132 213 111 223 121 222 232 313 323 221 312 311 333 231 322 321 233 131 212 211 133 332 331 Figure 1. 3-D Matrice. OALib Journal 2/11 O. Zaka 113 112 123 122 223 222 232 313 311 323 221 333 A = ( aijk )i =1,2,3 321 123 122 223 222 232 313 322 B = ( bijk )i =1,2,3 311 321 i =1,2,3 k =1,2,3 333 231 312 331 323 221 332 233 131 212 211 133 132 213 121 111 231 322 312 + 233 131 212 211 112 132 213 121 111 113 133 332 331 i =1,2,3 k =1,2,3 Figure 2. The addition of 3 × 3 × 3, 3D matrices. Definition 2.3 Zero matrix 3 × 3, 3D we will called the matrix that has all its elements zero. = O3×3×3 ) |i {( 0= F ijk a )|i {(= } 1,= 2, 3; j 1,= 2, 3; k 1, 2, 3 Definition 2.4. The opposite matric of anmatrice = A3×3×3 ijk } 1,= 2, 3; j 1,= 2, 3; k 1, 2, 3 { } − A3×3×3 = 1, 2, 3; j = 1, 2, 3; k = 1, 2, 3 ( −aijk ) | i = will, called matrix ( ) (where − aijk is a opposite element of element aijk ∈ F , so aijk + −aijk =0 F and ( F , +, ⋅ ) {( a + ( −a ))=| i 1, 2, 3;=j 1, 2, 3;=k 3; j 1, 2, = 3; k 1,= 2, 3} ) | i 1, 2,= {( 0= is field [8] [10] [11]), which satisfies the condition A3×3×3 + ( − A3×3×= 3) = ijk ijk } 1, 2, 3 O3×3×3 Theorem 2.1 ( 3×3×3 ( F ) , + ) is a beliangrup. Proof: Truly from the definition 2.2, of addition the 3-Dmatrices, we see that addition is the sustainable in 3×3×3 ( F ) , because ijk aijk ∈ F , bijk ∈ F ⇒ cijk = aijk + bijk ∈ F , ∀i, j , k ∈ {1, 2, 3} ∀A =( aijk ) , B =( bijk ) , C =( cijk ) ∈ 3×3×3 ( F ) ⇒ ( A + B ) + C =A + ( B + C ) 1) Associative property, ( A + B ) + C = ( aijk ) + ( bijk ) + ( cijk ) = ( aijk + bijk ) + ( cijk ) = ( ( aijk + bijk ) + cijk ) truly ( ) = ( aijk + bijk + cijk ) = aijk + ( bijk + cijk ) = ( aijk ) + ( bijk + cijk ) = A + ( B + C ) OALib Journal 3/11 ( ) ( ) O. Zaka ( ) ( ) 2) ∀A = aijk ∈ 3×3×3 ( F ) , ∃O = 0ijk / A + O = O + A = A. truly, ∀A = aijk ∈ 3×3×3 ( F ) , ∃O = 0ijk / A + O = O + A = A. {( a ) + ( 0) A += O { 3) ∀A = )|i {( a= = ijk } | i 1, 2, 3;= = j 1, 2, 3;= k 1, 2, 3 } = ( aijk + 0 ) | i =1, 2, 3; j =1, 2, 3; k =1, 2, 3 ijk } ijk 1, 2, 3; j 1, 2, 3; k 1,= 2, 3 A = = ( a ) ∈  ( F ) , ∃ − A = ( −a ) ∈  ( F ) / A + ( − A ) = ijk 3×3×3 ijk 3×3×3 {( a + ( −a ))=| i 1, 2, 3;=j 1, 2, 3;=k 3; j 1, 2, = 3; k 1,= 2, 3} ) | i 1, 2,= {( 0= ijk } truly, from Definition 2.4, we have A + ( −= A) = ijk O. 1, 2, 3 ijk O ijk ∀A = ( aijk ) , B = ( bijk ) ∈ 3×3×3 ( F ) , A + B = B + A. 4) Addition is commutative. truly A + B = ( aijk ) + ( bijk ) = ( aijk + bijk ) (  ,+ ) is abelian = (b ijk + aijk ) = ( bijk ) + ( aijk ) = B + A 3. Addition of m × n × p , 3-D Matrices over Any Field F and the Addition Abelian Group of Their 3-D Matrices Definition 3.1 3-dimensionalmxnxp matrix will call, a matrix which has: mhorizontal layers (analogous to m-rows), n-vertical page (analogue with n-columns in the usualmatrices) and p-vertical layers (p − 1 of which are hidden). { } m×n× p ( F ) = 1, m; j 1, n; k = 1, p ( aijk ) | aijk ∈ F -field and i == The set of these matrixes the write how: Definition 3.2 The addition of two matrices A, B ∈ m×n× p ( F ) we will call the matrix: C m×n× p= {( c ) | c = ijk } aijk + bijk , ∀= i 1, m; = j 1, n; k= 1, p ijk The appearance of the addition of mxnxp, 3D matrices will be as in Figure 3, )|i {( a = )|i {(b = where matrices A and B have the following appearance, Am= ×n× p Bm= ×n× p ijk ijk } 1, p} 1, = m; j 1,= n; k 1, p 1, = m; j 1,= n; k Definition 3.3 3-D, Zero matrix m × n × p , we will called the matrix that has all its elements zero. = O Om= ×n× p )|i {( a = | i {( 0) = ijk Definition 3.4 The opposite matric of anmatrice Am×= n× p ijk } 1, = m; j 1,= n; k 1, p } 1, m= ; j 1, n= ; k 1, p ∈ m×n× p ( F ) will, called matrix OALib Journal 4/11 O. Zaka 11p 112 22p 213 2n2 222 m1p 211 m2p mnp + m22 A = ( aijk )i =1,m m21 1n2 122 22p 213 2n2 222 212 m1p mn2 m22 m12 mn1 m2p mn2 B = ( bijk )i =1,m m11 mn1 m21 i =1, n k =1, p mnp 231 2n1 221 211 2np 131 1n1 121 111 231 2n1 221 m12 m11 2np 1np 12p 112 131 1n1 212 11p 1n2 122 121 111 1np 12p i =1, n k =1, p Figure 3. The addition of mxnxp, 3D matrices. { } − Am×n× p = 1, m; j = 1, n; k = 1, p ∈ m×n× p ( F ) ( −aijk ) | i = ( ) (where − aijk is a opposite element of element aijk ∈ F , so aijk + −aijk =0 F and ( F , +, ⋅) is field), which satisfies the condition {( a + ( −a ))=| i ; j {( 0) =| i 1, m= Am×n× p + ( − Am×n×= p) ijk = ijk } 1, m= ; j 1, n= ; k 1, p } n; k 1,= p O 1, = Theorem 3.1 ( m×n× p ( F ) , + ) is abeliangrup. Proof: Truly from the definition 3.2, of additions the 3-D matrices, we see that addition is the sustainable in m×n× p ( F ) , because ijk aijk ∈ F , bijk ∈ F ⇒ cijk= aijk + bijk ∈ F , ∀= i 1, m; = j 1, n; k= 1, p ∀A =( aijk ) , B =( bijk ) , C =( cijk ) ∈ m×n× p ( F ) ⇒ ( A + B ) + C =A + ( B + C ) 1) Associative property, ( A + B ) + C = ( aijk ) + ( bijk ) + ( cijk ) = ( aijk + bijk ) + ( cijk ) = ( ( aijk + bijk ) + cijk ) truly ( ) = ( aijk + bijk + cijk ) = aijk + ( bijk + cijk ) = ( aijk ) + ( bijk + cijk ) = A + ( B + C ) ( ) ( ) 2) ∀A = aijk ∈ m×n× p ( F ) , ∃O = 0ijk ∈ m×n× p ( F ) / A + O = O + A = A. ∀A = ( aijk ) ∈ m×n× p ( F ) , ∃O = ( 0ijk ) ∈ m×n× p ( F ) / A + O = O + A = A. truly, A+O = {( a ) + ( 0) { = OALib Journal )|i {( a = ijk ijk ijk } |= i 1, m; = j 1, n; = k 1, p } =( aijk + 0 F ) | i = 1, m; j = 1, n; k = 1, p ijk } 1, m= ; j 1, = n; k 1,= p A 5/11 3) ∀A = (a ) ∈  ijk m×n× p ( −a ) ∈  (F ),∃ − A = = m×n× p ( F ) / A + ( − A) = ; j 1, n= ;k {( a + ( −a ))=| i 1, m= n; k 1,= p} ; j 1, = {( 0 ) =| i 1, m= ijk truly, from Definition 2.4, we have A + ( −= A) O. Zaka ijk ijk F ijk } O. 1, p O ∀A = ( aijk ) , B = ( bijk ) ∈ m×n× p ( F ) / A + B = B + A. 4) Addition is commutative. truly A + B = ( aijk ) + ( bijk ) = ( aijk + bijk ) ( F ,+ ) is abelian = (b ijk + aijk ) = ( bijk ) + ( aijk ) = B + A 4. The “Common” Multiplication of 3 × 3 × 3 , 3-D Matrices with Elements Froman Field F Definition 4.1: The multiplication of two matrices A, B ∈ 3×3×3 ( F ) we will call the matrix C = A ⊗ B ∈ 3×3×3 ( F ) calculated as follows:  a113  3-vertical layer  a213   a313   a  112 ∀ 2-vertical layer  a212   a312     a111  1-vertical layer  a211 a  311 a123 a223 a323 a122 a222 a322 a121 a221 a321 a133   b113 b123 b133     a233  3-vertical layer  b213 b223 b233     a333   b313 b323 b333        b  b b a132    112 122 132  a232  , 2-vertical layer  b212 b222 b232  ∈ 3×3×3 ( F )     b312 b322 b332  a332           a131   b111 b121 b131     a231  1-vertical layer  b211 b221 b231  b  a331   311 b321 b331  The appearance of the multiplication of 3 × 3 × 3, 3D matrices will be as in Figure 4.  c113 c123 c133  ( 3-vertical layer )  c213 c223 c233   c313 c323 c333      c  c c  112 122 132  = C (= 2-vertical layer )  c212 c222 c232  c   312 c322 c332      c c c  111 121 131  (1-vertical layer )  c211 c221 c231  c   311 c321 c331   a113   a213  a313   a  112  a212 a  312    a111 a  211  a311 a123 a223 a323 a122 a222 a322 a121 a221 a321 a133   b113 b123   a233   b213 b223 a333   b313 b323     a132   b112 b122   a232  ⊗  b212 b222 a332   b312 b322     a131   b111 b121 a231   b211 b221   a331   b311 b321 b133   b233  b333    b132   b232  b332    b131  b231   b331  where, the first vertical page is: OALib Journal 6/11 O. Zaka 113 112 213 111 212 313 211 312 311 113 112 123 122 223 222 232 313 311 323 221 333 231 322 312 233 131 212 211 112 132 213 121 111 A = ( aijk )i =1,2,3 321 113 133 ⊗ 122 222 311 233 232 313 323 221 333 231 322 312 i =1,2,3 k =1,2,3 223 131 212 211 133 132 213 121 111 332 331 123 B = ( bijk )i =1,2,3 321 332 331 i =1,2,3 k =1,2,3 Figure 4. The multiplication of 3 × 3 × 3, 3D matrices. c111 = a111 ⋅ b111 + a121 ⋅ b211 + a131 ⋅ b311 ; c112 = a112 ⋅ b112 + a122 ⋅ b211 + a132 ⋅ b312 ; c113 = a113 ⋅ b113 + a123 ⋅ b213 + a133 ⋅ b313 ; c211 = a211 ⋅ b111 + a221 ⋅ b211 + a231 ⋅ b311 ; c212 = a212 ⋅ b112 + a222 ⋅ b211 + a232 ⋅ b312 ; c213 = a213 ⋅ b113 + a223 ⋅ b213 + a233 ⋅ b313 ; c311 = a311 ⋅ b111 + a321 ⋅ b211 + a331 ⋅ b311 ; c312 = a312 ⋅ b112 + a322 ⋅ b211 + a332 ⋅ b312 ; c313 = a313 ⋅ b113 + a323 ⋅ b213 + a333 ⋅ b313 ; the second vertical page is: c121 = a111 ⋅ b121 + a121 ⋅ b221 + a131 ⋅ b321 ; c122 = a112 ⋅ b122 + a122 ⋅ b222 + a132 ⋅ b322 ; c123 = a113 ⋅ b123 + a123 ⋅ b223 + a133 ⋅ b323 ; OALib Journal 7/11 O. Zaka c221 = a211 ⋅ b121 + a221 ⋅ b221 + a231 ⋅ b321 ; c222 = a212 ⋅ b122 + a222 ⋅ b222 + a232 ⋅ b322 ; c223 = a213 ⋅ b123 + a223 ⋅ b223 + a233 ⋅ b323 ; c321 = a311 ⋅ b121 + a321 ⋅ b221 + a331 ⋅ b321 ; c322 = a312 ⋅ b122 + a322 ⋅ b222 + a332 ⋅ b322 ; c323 = a313 ⋅ b123 + a323 ⋅ b223 + a333 ⋅ b323 ; and third vertical page is: c131 = a111 ⋅ b131 + a121 ⋅ b231 + a131 ⋅ b331 ; c132 = a112 ⋅ b132 + a122 ⋅ b232 + a132 ⋅ b332 ; c133 = a113 ⋅ b133 + a123 ⋅ b233 + a133 ⋅ b333 ; c231 = a211 ⋅ b121 + a221 ⋅ b221 + a231 ⋅ b321 ; c232 = a212 ⋅ b122 + a222 ⋅ b222 + a232 ⋅ b322 ; c233 = a213 ⋅ b123 + a223 ⋅ b223 + a233 ⋅ b323 ; c331 = a311 ⋅ b121 + a321 ⋅ b221 + a331 ⋅ b321 ; c332 = a312 ⋅ b122 + a322 ⋅ b222 + a332 ⋅ b322 ; c333 = a313 ⋅ b123 + a323 ⋅ b223 + a333 ⋅ b323 ; It is reduce the above notes through matrix blocks  C3   A3   B3   A3 × B3           C 2  = A2  ⊗  B2  = A2 × B2  C   A   B   A ×B  1   1  1  1  1 where  a111 a121 a131    = A1 = a211 a221 a231  ; A2 a   311 a321 a331   b111 b121 b131    B1 = = b211 b221 b231  ; B2 b   311 b321 b331   a112 a122 a132   a113 a123 a133      a212 a222 a232  ; A3  a213 a223 a233  ; = a  a   312 a322 a332   313 a323 a333   b112 b122 b132   b113 b123 b133      b212 b222 b232  ; B3  b213 b223 b233  ; = b  b   312 b322 b332   313 b323 b333   C3   A3   B3   A3 × B3           C 2  = A2  ⊗  B2  = A2 × B2  C   A   B   A ×B  1   1  1  1  1 and  c111 c121 c131    C1 = = c211 c221 c231  ; C 2 c   311 c321 c331   c112 c122 c132    c212 c222 c232  ; C3 = c   312 c322 c332   c113   c213 c  313 C1 =× A1 B1 ; C 2 =× A2 B2 ; C3 =× A3 B3 c123 c223 c323 c133   c233  c333  Remark 4.1 Two dimensional matrices can think like matrix with size m × n ×1 Easy seen from the definition 1, above it that, if = aij 2 0,= aij 3 0 and = bij 2 0,= bij 3 0, ∀i, j ∈ (1, 2, 3) we get, the usual 3 × 3-matrix multiplication, then will take only the first vertical layer is (or, in the language of matrix blocks OALib Journal 8/11 O. Zaka would say that: A2 = 0 ; A3 = 0 ; B2 = 0 ; B3 = 0 ):  a111 ⋅ b111 + a121 ⋅ b211 + a131 ⋅ b311   a211 ⋅ b111 + a221 ⋅ b211 + a231 ⋅ b311  a ⋅b + a ⋅b + a ⋅b 311 321 211 331  311 111 a111 ⋅ b121 + a121 ⋅ b221 + a131 ⋅ b321 a211 ⋅ b121 + a221 ⋅ b221 + a231 ⋅ b321 a311 ⋅ b121 + a321 ⋅ b221 + a331 ⋅ b321 a111 ⋅ b131 + a121 ⋅ b231 + a131 ⋅ b331   a211 ⋅ b121 + a221 ⋅ b221 + a231 ⋅ b321  a311 ⋅ b121 + a321 ⋅ b221 + a331 ⋅ b321  Definition 4.2. The3-D,unit matrix, associated with the “common” multiplication, must be: I 3×3×3 1  0 0   1  = 0 0    1 0  0  0 0  1 0 third vertical layer 0 1   0 0  1 0  the second vertical layer 0 1    0 0 1 0  the first vertical layer  0 1  or, in the language of matrix blocks: I 3×3×3  I 3×3    =  I 3×3  I   3×3  Easy distinguish that, ∀A ∈ 3×3×3 ( F ) / A ⊗ I 3×3×3 =A. Theorem 4.1 ( 3×3×3 ( F ) , ⊗ ) is a unitary semi-Group with regard to this ordinary multiplication Proof: 1) associative property. ∀A, B, C ∈ 3×3×3 ( F )  A3   B3    C3   A3 × B3   C3   ( A3 × B3 ) × C3                A2  ⊗  B2   ⊗  C 2  = A2 × B2  ⊗  C 2  = ( A2 × B2 ) × C2   A1   B1    C1   A1 × B1   C1   ( A1 × B1 ) × C1     A3 × ( B3 × C3 )   C2 )   A2 × ( B2 ×=  A ×(B ×C )  1 1   1 ( 3×3 ( F ),×) is a semigroup  =  A3   B3   C3           A2  ⊗  B2  ⊗  C 2   .  A   B   C    1   1   1   2) ∃I 3×3×3 ∈ 3×3×3 ( F ) / ∀A ∈ 3×3×3 ( F ) ⇒ A × I 3×3×3 =A.  A3   I 3×3   A3 × I 3×3  (  ( F ),×) is a unitary semigroup  A3     3×3      =  A2 × I 3×3   A2  ⊗  I 3×3  =  A2           A1   I 3×3   A1 × I 3×3   A1  Theorem 4.2 ( 3×3×3 ( F ) , +, ⊗ ) is a unitary Ring. Proof: 1) From Theorem 2.1. ( 3×3×3 ( F ) , + ) is abeliangrup. 2) From Theorem 4.1. ( 3×3×3 ( F ) , ⊗ ) is a unitary semi-Group, and consequently also, ( 3×3×3 ( F ) , ⊗ ) is a unitary semi-Group 3) ∀A, B, C ∈ 3×3×3 ( F ) , a) A ⊗ ( B + C ) = A ⊗ B + A ⊗ C . b) ( A + B ) ⊗ C = A ⊗ C + B ⊗ C . OALib Journal 9/11 O. Zaka truly  A3   B3   C3          A ⊗ ( B + C )=  A2  ⊗  B2  +  C 2  =  A   B   C    1   1   1    A3   B3 + C3       A2  ⊗  B2 + C 2 =  A   B +C  1   1  1  A3 × ( B3 + C3 )     A2 × ( B2 + C 2 )   A ×(B + C )  1 1   1 A3 × B3 + A3 × C3   A3 × B3   A3 × C3        =  A2 × B2 + A2 × C 2  =  A2 × B2  +  A2 × C 2         A1 × B1 + A1 × C1   A1 × B1   A1 × C1    A3   B3     A3   C3           =   A2  ⊗  B2   +   A2  ⊗  C 2   = A ⊗ B + A ⊗ C .  A   B   A   C   1   1   1   1  ( 3×3 ( F ),+ ,×) is a unitary Ring  In a similar manner proved the point (b). 5. Multiplication of a 3-D, 3 × 3 × 3 -Matrix by a Scalar Definition 5.1 The multiplication of matrix A ∈ 3×3×3 ( F ) with scalar λ ∈ F , is matrix= C λ  A ∈ 3×3×3 ( F ) :  a113 a123 a133     a213 a223 a233   a313 a323 a333      a a122 a132  112   = C λ=   a212 a222 a232  a   312 a322 a332       a111 a121 a131  a a221 a231   211   a311 a321 a331   λ ⋅ a113   λ ⋅ a213  λ ⋅ a313   λ ⋅a 112   λ ⋅ a212 λ ⋅a 312     λ ⋅ a111 λ ⋅a 211  λ a ⋅ 311  λ ⋅ a123 λ ⋅ a133  λ ⋅ a223 λ ⋅ a233  λ ⋅ a323 λ ⋅ a333  λ ⋅ a122 λ ⋅ a222 λ ⋅ a322 λ ⋅ a121 λ ⋅ a221 λ ⋅ a321   λ ⋅ a132   λ ⋅ a232  λ ⋅ a332    λ ⋅ a131  λ ⋅ a231   λ ⋅ a331   : F × 3×3×3 ( F ) → 3×3×3 ( F ) So ( λ , A)  λ  A Theorem 5.1 ( 3×3×3 ( F ) , +,  F ) is a vector space Proof. is evident because, ( 3×3 ( F ) , +,  F ) it is the vector space, see [6] [8] Definition 5.2 The multiplication of matrix A ∈ m×n× p ( F ) with scalar λ ∈ F , is matrix= C λ  A ∈ m×n× p ( F ) : [9] [10]. wherein each element of the matrix is multiplied (by multiplication of the field F) with the element λ ∈ F . Well, so we have  : F × m×n× p ( F ) → m×n× p ( F ) ( F ) , +,  F ) is a vector space Proof. Is evident because, ( m×n ( F ) , +,  F ) it is the vector space, see [6] [8] Theorem 5.2 ( ( λ , A)  λ  A m×n× p [9] [10] OALib Journal 10/11 O. Zaka 6. Conclusion In this article, based on geometric considerations, and mostly considering the cube, we managed to develop the idea of the 3D matrix doing so a generalization of the 2D matrices, step by step. Furthermore, we gave a unitary ring with the elements of a field F. Initially we gave the ring 3 × 3 × 3 , 3D matrices and then generalized this concept for m × n × p , 3D matrices. At the end of this article, we present the scalar multiplication with the 3D matrices and we show that the set of 3 × 3 × 3 , 3D matrix, forms a vector space over the field F. References [1] Artin, M. (1991) Algebra. Prentice Hall, Upper Saddle River. [2] Bretscher, O. (2005) Linear Algebra with Applications. 3rd Edition, Prentice Hall, Upper Saddle River. [3] Connell, E.H. (2004) Elements of Abstract and Linear Algebra. [4] Schneide, H. and Barker, G.P. (1973) Matrices and Linear Algebra (Dover Books on Mathematics). 2nd Revised Edition. [5] Lang, S. (1987) Linear Algebra. Springer-Verlag, Berlin, New York. [6] Nering, E.D. (1970) Linear Algebra and Matrix Theory. 2nd Edition, Wiley, New York. [7] Zaka, O. and Filipi, K. (2016) The Transform of a Line of Desargues Affine Plane in an Additive Group of Its Points. International Journal of Current Research, 8, 34983-34990. [8] Zaka, O. (2013) Abstract Algebra II. Vllamasi, Tirana. [9] Zaka, O. (2013) Linear Algebra I. Vllamasi, Tirana. [10] Zaka, O. (2013) Linear Algebra II. Vllamasi, Tirana. [11] Zaka, O. and Filipi, K. (2016) One Construction of an Affine Plane over a Corps. Journal of Advances in Mathematics, 12. 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