Tauira o ngā Pātai Kōrero mō te Whakareatanga Matrix
Ko te whakarea matihiko he ariā taketake i roto i te taurangi rārangi e whakamahia pinepinetia ana i roto i ngā momo mara pēnei i te ahupūngao, te whakairoiro rorohiko, me te ako mīhini. I roto i tēnei tuhinga, ka matapakihia e mātou ngā ariā taketake o te whakarea matihiko, te "ture tāpiri huānga-ā-ira," me te whakarato hoki i ētahi tauira rapanga me ō rātou otinga.
Ngā Ariā Taketake o te Whakareatanga Matrix
I mua i te titiro ki ngā tauira rapanga, he mea nui kia mārama ki ngā ture taketake o te whakareatanga matihiko. Me kī he rua ā tātou matihiko \( A \) me \( B \) kei reira:
– He rahi te matihiko \( A \) \( m \times n \)
– He rahi te matihiko \( B \) \( n \times p \)
Hei whakarea i ngā matihiko e rua \( A \) me \( B \), me ōrite te maha o ngā pou o te matihiko \( A \) ki te maha o ngā rarangi o te matihiko \( B \) (arā, e rua \( n \)). Ko te hua o ēnei matihiko he matihiko \( C \) he rahi \( m \times p \) kei reira ngā huānga \( C_{ij} \) e tautuhia ana ko:
\[ C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} \]
Ko te tikanga o tēnei ko ia huānga o te matihiko hua ko te tapeke o ngā hua o ngā huānga o te rarangi \( i \) o te matihiko \( A \) me ngā huānga o te pou \( j \) o te matihiko \( B \).
Ngā Pātai Tauira me te Kōrero
Pātai 1: Te Whakarea o ngā Matriki 2×2
Me kī kei a tātou ngā matihiko \( A \) me \( B \) penei:
\[ A = \begin{pmatrix} 1 me te 2 \\ 3 me te 4 \end{pmatrix} \]
\[ B = \begin{pmatrix} 2 me te 0 \\ 1 me te 3 \end{pmatrix} \]
Whakareatia ngā matihiko \( A \) me \( B \) kia whiwhi ai i te matihiko \( C \).
Kōrero:
Me tatau tātou i ngā huānga o te matihiko \( C \):
\[ C_{11} = 1 \cdot 2 + 2 \cdot 1 = 2 + 2 = 4 \]
\[ C_{12} = 1 \cdot 0 + 2 \cdot 3 = 0 + 6 = 6 \]
\[ C_{21} = 3 \cdot 2 + 4 \cdot 1 = 6 + 4 = 10 \]
\[ C_{22} = 3 \cdot 0 + 4 \cdot 3 = 0 + 12 = 12 \]
Nō reira, ko te hua o te matihiko \( C \) ko:
\[ C = \begin{pmatrix} 4 me te 6 \\ 10 me te 12 \end{pmatrix} \]
Pātai 2: Te Whakarea o ngā Matriki 3×3
Me kī kei a tātou ngā matihiko \( D \) me \( E \) penei:
\[ D = \begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 0 \end{pmatrix} \]
\[ E = \begin{pmatrix} 3 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix} \]
Whakareatia ngā matihiko \( D \) me \( E \) kia whiwhi ai i te matihiko \( F \).
Kōrero:
Me tatau tātou i ngā huānga o te matihiko \( F \):
\[ F_{11} = 1 \cdot 3 + 0 \cdot 2 + 2 \cdot 1 = 3 + 0 + 2 = 5 \]
\[ F_{12} = 1 \cdot 1 + 0 \cdot 1 + 2 \cdot 0 = 1 + 0 + 0 = 1 \]
\[ F_{13} = 1 \cdot 2 + 0 \cdot 1 + 2 \cdot 1 = 2 + 0 + 2 = 4 \]
\[ F_{21} = -1 \cdot 3 + 3 \cdot 2 + 1 \cdot 1 = -3 + 6 + 1 = 4 \]
\[ F_{22} = -1 \cdot 1 + 3 \cdot 1 + 1 \cdot 0 = -1 + 3 + 0 = 2 \]
\[ F_{23} = -1 \cdot 2 + 3 \cdot 1 + 1 \cdot 1 = -2 + 3 + 1 = 2 \]
\[ F_{31} = 2 \cdot 3 + 1 \cdot 2 + 0 \cdot 1 = 6 + 2 + 0 = 8 \]
\[ F_{32} = 2 \cdot 1 + 1 \cdot 1 + 0 \cdot 0 = 2 + 1 + 0 = 3 \]
\[ F_{33} = 2 \cdot 2 + 1 \cdot 1 + 0 \cdot 1 = 4 + 1 + 0 = 5 \]
Nō reira, ko te hua o te matihiko \( F \) ko:
\[ F = \begin{pmatrix} 5 me te 1 me te 4 \\ 4 me te 2 me te 2 \\ 8 me te 3 me te 5 \end{pmatrix} \]
Pātai 3: Te whakarea o tētahi Matū 2×3 ki tētahi Matū 3×2
Me kī kei a tātou ngā matihiko \( G \) me \( H \) penei:
\[ G = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \]
\[ H = \begin{pmatrix} 7 me te 8 \\ 9 me te 10 \\ 11 me te 12 \end{pmatrix} \]
Whakareatia ngā matihiko \( G \) me \( H \) kia whiwhi ai i te matihiko \( I \).
Kōrero:
Me tatau tātou i ngā huānga o te matihiko \( I \):
\[ I_{11} = 1 \cdot 7 + 2 \cdot 9 + 3 \cdot 11 = 7 + 18 + 33 = 58 \]
\[ I_{12} = 1 \cdot 8 + 2 \cdot 10 + 3 \cdot 12 = 8 + 20 + 36 = 64 \]
\[ I_{21} = 4 \cdot 7 + 5 \cdot 9 + 6 \cdot 11 = 28 + 45 + 66 = 139 \]
\[ I_{22} = 4 \cdot 8 + 5 \cdot 10 + 6 \cdot 12 = 32 + 50 + 72 = 154 \]
Nō reira, ko te hua o te matihiko \( I \) ko:
\[ I = \begin{pmatrix} 58 me te 64 \\ 139 me te 154 \end{pmatrix} \]
Whakamutunga
I roto i tēnei tuhinga, kua hipokina e mātou ngā ture taketake o te whakarea matihiko, ā, kua whakaratohia e mātou ētahi tauira raruraru e toru me ngā whakamārama. He pūnaha te tukanga tatau i te whakarea matihiko, e hiahia ana kia aro nui ki ngā whakarea o ia huānga matihiko me ō rātou tapeke. Mā te mārama me te mahi auau i ngā raruraru whakarea matihiko, ka pai ake te mārama ki tēnei ariā, ā, ka taea hoki te whakamahi i roto i ngā momo mara pūtaiao.
Ehara i te mea he tūāpapa nui te whakarea matihiko i roto i te pāngarau me te pūtaiao rorohiko anake, engari he tino whai hua hoki i roto i ngā tono o te ao tūturu, pērā i te tātari raraunga, te arotau, tae atu ki ngā rauropi ako mīhini. Nō reira, he tūāpapa nui te māramatanga pai ki te whakarea matihiko mō tētahi tohunga pāngarau, mō tētahi kaipūtaiao rorohiko rānei.
