Ngā Tauira Pātai e Matapaki ana i ngā Whakatau me ngā Whakamuri o te Matrix
Ko ngā whakatau matihiko me ngā whakahurihuri matihiko he ariā matua e rua i roto i te arapūrangi rārangi e whānuitia ana te whakamahinga i roto i ngā mara maha, tae atu ki te pāngarau, te ahupūngao, te ōhanga, me te hangarau. He mea nui te māramatanga hōhonu ki ēnei ariā hei whakaoti rapanga pāngarau uaua. I roto i tēnei tuhinga, ka matapakihia e mātou ngā tauira o ngā whakatau matihiko me ngā whakahurihuri, me tētahi matapakinga whānui.
Kaiwhakatau Matrix
Ko te whakatau he tauine e pā ana ki tētahi matihiko tapawhā (he matihiko he rite te maha o ngā rarangi me ngā pou). Ka taea e te whakatau te whakarato i ngā mōhiohio nui mō ngā āhuatanga o te matihiko, pēnei i te mea ka taea te huri, te kore rānei.
Tauira Pātai 1: Kaiwhakatau o tētahi Matrix 2×2
E whai ake nei te āhua o te matihiko \( A \):
\[
A = \begin{pmatrix}
4 me te 3
2 & 1
\end{pmatrix}
\]
Whakatauhia te kaiwhakatau o te matihiko \( A \).
Kōrero:
Mō tētahi matihiko 2×2, ka taea te tatau i te whakatau mā te whakamahi i te tātai māmā e whai ake nei:
\[
\text{det}(A) = ad – bc
\]
kei hea \( A = \begin{pmatrix} a me b \\ c me d \end{pmatrix} \).
Te whakakapinga o ngā huānga o te matihiko \( A \):
\[
\text{det}(A) = (4 \times 1) – (3 \times 2) = 4 – 6 = -2
\]
Nō reira, ko te whakatau o te matihiko \( A \) ko -2.
Tauira Pātai 2: Kaiwhakatau o tētahi Matrix 3×3
E whai ake nei te āhua o te matihiko \( B \):
\[
B = \begin{pmatrix}
1 me te 2 me te 3
0 me te 1 me te 4
5 & 6 & 0
\end{pmatrix}
\]
Whakatauhia te kaiwhakatau o te matihiko \( B \).
Kōrero:
Mō tētahi matihiko 3×3, ka taea te tatau i te whakatau mā te whakamahi i te ture a Sarrus, i ngā tauwehenga rānei. I konei, ka whakamahia e tātou te ture a Sarrus hei whakahaere i te tatau.
Tāruatia ngā pou tuatahi e rua ki te taha matau o te matihiko:
\[
\text{det}(B) = \begin{vmatrix}
1 me te 2 me te 3
0 me te 1 me te 4
5 & 6 & 0
\end{vmatrix}
= 1\cdot1\cdot0 + 2\cdot4\cdot5 + 3\cdot0\cdot6 – (3\cdot1\cdot5 + 2\cdot0\cdot0 + 1\cdot4\cdot6)
\]
\[
= 0 + 40 + 0 – (15 + 0 + 24)
\]
\[
= 40 – 39 = 1
\]
Nō reira, ko te whakatau o te matihiko \( B \) ko 1.
Matrix Whakamuri
Ko te whakahuri o tētahi matihiko \( A \) (mēnā kei te wātea) he matihiko \( A^{-1} \) e tutuki ana i ngā tikanga e whai ake nei:
\[
A \cdot A^{-1} = A^{-1} \cdot A = I
\]
ko \( I \) te matihiko tuakiri e 1 ana ngā huānga whakarara, ā, ko 0 ngā huānga kē atu.
Tauira Pātai 3: Te Hurihanga o te Matrix 2×2
E whai ake nei te āhua o te matihiko \( C \):
\[
C = \begin{pmatrix}
1 me te 2
3 & 4
\end{pmatrix}
\]
Kimihia te whakahuri o te matihiko \( C \).
Kōrero:
Mō tētahi matihiko 2×2, ka taea te tatau i te whakahurihanga mā te whakamahi i te tātai:
\[
C^{-1} = \frac{1}{\text{det}(C)} \begin{pmatrix}
d & -b \\
-c me te a
\end{pmatrix}
\]
kei hea \( C = \begin{pmatrix} a me b \\ c me d \end{pmatrix} \).
Tuatahi, ka tatauhia e mātou te kaiwhakatau o te matihiko \( C \):
\[
\text{det}(C) = (1 \cdot 4) – (2 \cdot 3) = 4 – 6 = -2
\]
Kātahi, whakakapia ki te tātai whakamuri:
\[
C^{-1} = \frac{1}{-2} \begin{pmatrix}
4 me te -2
-3 me te 1
\end{pmatrix}
= \begin{pmatrix}
-2 me te 1
\frac{3}{2} me -\frac{1}{2}
\end{pmatrix}
\]
Nō reira, ko te whakahurihanga o te matihiko \( C \) ko \( \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \).
Tauira Pātai 4: Te Hurihanga o te Matrix 3×3
E whai ake nei te āhua o te matihiko \( D \):
\[
D = \begin{pmatrix}
2 me te 0 me te 1
3 me te 0 me te 0
1 & 4 & 2
\end{pmatrix}
\]
Kimihia te whakahuri o te matihiko \( D \).
Kōrero:
Mō ngā matihiko 3×3, n×n rānei, ko te tikanga e whakamahia whānuitia ana ko te tikanga echelon, ko te tikanga adjoint rānei. I konei, ka whakamahia e tātou te tikanga echelon.
Ko te taahiraa tuatahi ko te hanga i te matihiko whakanuia \( [D|I] \) ko \( I \) te matihiko tuakiri:
\[
\left[\begin{array}{ccc|ccc}
2 me te 0 me te 1 me te 1 me te 0 me te 0
3 me te 0 me te 0 me te 0 me te 1 me te 0
1 me te 4 me te 2 me te 0 me te 0 me te 1
\end{array}\right]
\]
Kātahi, mahihia ngā mahi rarangi taketake kia hangaia rā anō te matihiko tuakiri i te taha maui:
1. Rārangi 1: \( B_1 \div 2 \)
\[
\left[\begin{array}{ccc|ccc}
1 me te 0 me te \frac{1}{2} me te \frac{1}{2} me te 0 me te 0 \\
3 me te 0 me te 0 me te 0 me te 1 me te 0
1 me te 4 me te 2 me te 0 me te 0 me te 1
\end{array}\right]
\]
2. Rārangi 2: \( B_2 – 3B_1 \)
\[
\left[\begin{array}{ccc|ccc}
1 me te 0 me te \frac{1}{2} me te \frac{1}{2} me te 0 me te 0 \\
0 me te 0 me te -\frac{3}{2} me te -\frac{3}{2} me te 1 me te 0
1 me te 4 me te 2 me te 0 me te 0 me te 1
\end{array}\right]
\]
3. Rārangi 3: \( B_3 – B_1 \)
\[
\left[\begin{array}{ccc|ccc}
1 me te 0 me te \frac{1}{2} me te \frac{1}{2} me te 0 me te 0 \\
0 me te 0 me te -\frac{3}{2} me te -\frac{3}{2} me te 1 me te 0
0 me te 4 me te \frac{3}{2} me te -\frac{1}{2} me te 0 me te 1
\end{array}\right]
\]
4. Rārangi 3: \( B_3 \div 4 \)
\[
\left[\begin{array}{ccc|ccc}
1 me te 0 me te \frac{1}{2} me te \frac{1}{2} me te 0 me te 0 \\
0 me te 0 me te -\frac{3}{2} me te -\frac{3}{2} me te 1 me te 0
0 me te 1 me te \frac{3}{8} me te -\frac{1}{8} me te 0 me te \frac{1}{4}
\end{array}\right]
\]
5. Rārangi 1: \( B_1 – \frac{1}{2}B_3 \)
\[
\left[\begin{array}{ccc|ccc}
1 me te 0 me te 0 me te \frac{5}{16} me te 0 me te -\frac{1}{8} \\
0 me te 0 me te -\frac{3}{2} me te -\frac{3}{2} me te 1 me te 0
0 me te 1 me te \frac{3}{8} me te -\frac{1}{8} me te 0 me te \frac{1}{4}
\end{array}\right]
\]
6. Rārangi 2: \( B_2 \div -\frac{3}{2} \)
\[
\left[\begin{array}{ccc|ccc}
1 me te 0 me te 0 me te \frac{5}{16} me te 0 me te -\frac{1}{8} \\
0 me te 0 me te 1 me te 1 me te -\frac{2}{3} me te 0 \\
0 me te 1 me te \frac{3}{8} me te -\frac{1}{8} me te 0 me te \frac{1}{4}
\end{array}\right]
\]
7. Rārangi 3: \( B_3 – \frac{3}{8} B_2 \)
\[
\left[\begin{array}{ccc|ccc}
1 me te 0 me te 0 me te \frac{5}{16} me te 0 me te -\frac{1}{8} \\
0 me te 0 me te 1 me te 1 me te -\frac{2}{3} me te 0 \\
0 me te 1 me te 0 me te -\frac{1}{4} me te \frac{1}{6} me te \frac{1}{4}
\end{array}\right]
\]
Nō reira, ko te whakahurihanga o te matihiko \( D \) ko \( \begin{pmatrix} \frac{5}{16} & 0 & -\frac{1}{8} \\ 1 & -\frac{2}{3} & 0 \\ -\frac{1}{4} & \frac{1}{6} & \frac{1}{4} \end{pmatrix} \).
Mā te mārama ki ngā ariā me ngā tauira tuturu, ka kitea e tātou ka taea te tatau i ngā whakatau me ngā whakahurihuri o ngā matihiko mā te whakamahi i ngā tikanga māmā noa iho, engari he pānga nui ki te tātari raraunga me te whakaoti rapanga pāngarau uaua ake. He mea nui tēnei māramatanga i roto i ngā momo tono, tae atu ki ngā whakairoiro rorohiko, te tātari raraunga, me ngā pūnaha whārite rārangi.
