Mohlala oa Lipotso tsa Puisano tsa Katiso ea Matrix
Ho atisa matrix ke mohopolo oa motheo ho algebra e otlolohileng e sebelisoang khafetsa mafapheng a fapaneng a kang fisiks, litšoantšo tsa khomphutha le ho ithuta ka mochini. Sehloohong sena, re tla tšohla likhopolo tsa motheo tsa ho atisa matrix, "molao oa ho eketsa likarolo," 'me re tla boela re fane ka mehlala e' maloa ea mathata le litharollo tsa 'ona.
Mehopolo ea Motheo ea ho Atisa Matrix
Pele o sheba mehlala ea mathata, ho bohlokoa ho utloisisa melao ea motheo ea katiso ea matrix. A re re re na le matrices tse peli \( A \) le \( B \) moo:
– Matrix \( A \) e na le boholo \( m \times n \)
– Matrix \( B \) e na le boholo \( n \times p \)
Ho atisa matrices tse peli \( A \) le \( B \), palo ea likholomo tsa matrices \( A \) e tlameha ho lekana le palo ea mela ea matrices \( B \) (ke hore, ka bobeli \( n \)). Sehlahisoa sa matrices tsena ke matrices \( C \) ea boholo \( m \times p \) moo likarolo \( C_{ij} \) li hlalosoang e le:
\[ C_{ij} = \ kakaretso_{k=1}^{n} A_{ik} B_{kj} \]
Sena se bolela hore karolo e 'ngoe le e 'ngoe ea matrix e hlahisoang ke kakaretso ea lihlahisoa tsa likarolo tsa mola \( i \) oa matrix \( A \) le likarolo tsa kholomo \( j \) ea matrix \( B \).
Lipotso tsa Mehlala le Puisano
Potso ea 1: Katiso ea 2×2 Matrices
A re re re na le matrices \( A \) le \( B \) ka tsela e latelang:
\[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \]
\[ B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} \]
Atisa matrices \( A \) le \( B \) ho fumana matrices e hlahang \( C \).
Puisano:
A re baleng likarolo tsa matrix \( C \):
\[ C_{11} = 1 \cdot 2 + 2 \cdot 1 = 2 + 2 = 4 \]
\[ C_{12} = 1 \cdot 0 + 2 \cdot 3 = 0 + 6 = 6 \]
\[ C_{21} = 3 \cdot 2 + 4 \cdot 1 = 6 + 4 = 10 \]
\[ C_{22} = 3 \cdot 0 + 4 \cdot 3 = 0 + 12 = 12 \]
Kahoo, matrix e hlahang \( C \) ke:
\[ C = \begin{pmatrix} 4 & 6 \\ 10 & 12 \end{pmatrix} \]
Potso ea 2: Katiso ea 3×3 Matrices
A re re re na le matrices \( D \) le \( E \) ka tsela e latelang:
\[ D = \begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 0 \end{pmatrix} \]
\[ E = \begin{pmatrix} 3 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix} \]
Atisa matrices \( D \) le \( E \) ho fumana matrices e hlahang \( F \).
Puisano:
A re baleng likarolo tsa matrix \( F \):
\[ F_{11} = 1 \cdot 3 + 0 \cdot 2 + 2 \cdot 1 = 3 + 0 + 2 = 5 \]
\[ F_{12} = 1 \cdot 1 + 0 \cdot 1 + 2 \cdot 0 = 1 + 0 + 0 = 1 \]
\[ F_{13} = 1 \cdot 2 + 0 \cdot 1 + 2 \cdot 1 = 2 + 0 + 2 = 4 \]
\[ F_{21} = -1 \cdot 3 + 3 \cdot 2 + 1 \cdot 1 = -3 + 6 + 1 = 4 \]
\[ F_{22} = -1 \cdot 1 + 3 \cdot 1 + 1 \cdot 0 = -1 + 3 + 0 = 2 \]
\[ F_{23} = -1 \cdot 2 + 3 \cdot 1 + 1 \cdot 1 = -2 + 3 + 1 = 2 \]
\[ F_{31} = 2 \cdot 3 + 1 \cdot 2 + 0 \cdot 1 = 6 + 2 + 0 = 8 \]
\[ F_{32} = 2 \cdot 1 + 1 \cdot 1 + 0 \cdot 0 = 2 + 1 + 0 = 3 \]
\[ F_{33} = 2 \cdot 2 + 1 \cdot 1 + 0 \cdot 1 = 4 + 1 + 0 = 5 \]
Kahoo, matrix e hlahang \( F \) ke:
\[ F = \begin{pmatrix} 5 & 1 & 4 \\ 4 & 2 & 2 \\ 8 & 3 & 5 \end{pmatrix} \]
Potso ea 3: Katiso ea Matrix ea 2×3 ka Matrix ea 3×2
A re re re na le matrices \( G \) le \( H \) ka tsela e latelang:
\[ G = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \]
\[ H = \begin{pmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{pmatrix} \]
Atisa matrices \( G \) le \( H \) ho fumana matrices e hlahang \( I \).
Puisano:
A re baleng likarolo tsa matrix \( I \):
\[ I_{11} = 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 = 7 + 18 + 33 = 58 \]
\[ I_{12} = 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 = 8 + 20 + 36 = 64 \]
\[ I_{21} = 4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 = 28 + 45 + 66 = 139 \]
\[ I_{22} = 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12 = 32 + 50 + 72 = 154 \]
Kahoo, matrix e hlahang \( I \) ke:
\[ I = \begin{pmatrix} 58 & 64 \\ 139 & 154 \end{pmatrix} \]
Qetello
Sehloohong sena, re akarelitse melao ea motheo ea katiso ea matrix 'me ra fana ka mehlala e meraro ea mathata ka litlhaloso. Ts'ebetso ea ho bala katiso ea matrix e hlophisehile, e hlokang tlhokomelo e qaqileng ho li-multiplier tsa karolo e 'ngoe le e 'ngoe ea matrix le lipalo tsa tsona. Ka ho utloisisa le ho itloaetsa mathata a katiso ea matrix khafetsa, re tla utloisisa mohopolo ona hamolemo 'me re khone ho o sebelisa masimong a fapaneng a saense.
Katiso ea matrix ha se motheo oa bohlokoa feela lipalo le mahlale a khomphutha, empa hape e na le thuso haholo lits'ebetsong tsa lefats'e la nnete, joalo ka tlhahlobo ea data, ntlafatso, esita le li-algorithms tsa ho ithuta ka mochini. Ka hona, kutloisiso e ntle ea katiso ea matrix ke motheo oa bohlokoa bakeng sa setsebi leha e le sefe sa lipalo kapa rasaense oa khomphutha.
