Fa'ata'ita'iga o Fesili e Talanoaina ai Mea e Fa'atatau ma Fa'afeagai o Matrix
O matrix determinants ma matrix inverses o ni manatu autū se lua i le linear algebra e lautele lona faʻaaogaina i matāʻupu eseese, e aofia ai le matematika, fisiki, tamaoaiga, ma le inisinia. O se malamalamaga maeʻaeʻa i nei manatu e taua tele mo le foʻiaina o le tele o faʻafitauli faigata o le matematika. I totonu o lenei tusiga, o le a tatou talanoaina ai faʻataʻitaʻiga o matrix determinants ma inverses, faʻatasi ai ma se talanoaga atoatoa.
Fa'ailoga o le Matrix
O le determinant o se scalar e fesoʻotaʻi ma se square matrix (o se matrix e tutusa le aofaʻi o laina ma koluma). E mafai e le determinant ona tuʻuina atu faʻamatalaga taua e uiga i meatotino o le matrix, e pei o le mafai ona faʻaliliuina pe leai.
Fa'ata'ita'iga Fesili 1: Mea e Fa'atatau i se Matrix 2×2
I le tuuina atu o le matrix \( A \) e pei ona taua i lalo:
\[
A = \begin{pmatrix}
4 & 3 \\
2 & 1
\end{pmatrix}
\]
Fuafua le mea e fa'atatau i le matrix \( A \).
Talanoaga:
Mo se matrix 2×2, e mafai ona fuafuaina le determinant e faʻaaoga ai le fua faigofie lenei:
\[
\text{det}(A) = ad – bc
\]
lea \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).
Suiga o elemene o le matrix \( A \):
\[
\text{det}(A) = (4 \times 1) – (3 \times 2) = 4 – 6 = -2
\]
O lea la, o le determinant o le matrix \( A \) o le -2.
Fa'ata'ita'iga Fesili 2: Mea e Fa'atatau i se Matrix 3×3
I le tuuina atu o le matrix \( B \) e pei ona taua i lalo:
\[
B = \begin{pmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5&6&0
\end{pmatrix}
\]
Fuafua le mea e fa'atatau i le matrix \( B \).
Talanoaga:
Mo se matrix 3×3, e mafai ona fuafuaina le determinant e faʻaaoga ai le tulafono a Sarrus poʻo cofactors. Iinei, o le a tatou faʻaaogaina le tulafono a Sarrus e faʻafaigofie ai le fuafuaga.
Fa'alua koluma muamua e lua i le itu taumatau o le matrix:
\[
\text{det}(B) = \begin{vmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5&6&0
\end{vmatrix}
= 1\cdot1\cdot0 + 2\cdot4\cdot5 + 3\cdot0\cdot6 – (3\cdot1\cdot5 + 2\cdot0\cdot0 + 1\cdot4\cdot6)
\]
\[
= 0 + 40 + 0 – (15 + 0 + 24)
\]
\[
= 40 – 39 = 1
\]
O lea la, o le mea e fuafua ai le matrix \( B \) e 1.
Matrix Fa'afeagai
O le fa'afeagai o se matrix \( A \) (afai e iai) o se matrix \( A^{-1} \) e fa'amalieina ai tulaga nei:
\[
A \cdot A^{-1} = A^{-1} \cdot A = I
\]
lea o le \(I \) o le identity matrix o ona elemene diagonal e 1 ma isi elemene e 0.
Fa'ata'ita'iga Fesili 3: Fa'afeagai o se Matrix 2×2
I le tuuina atu o le matrix \( C \) e pei ona taua i lalo:
\[
C = \begin{pmatrix}
1 & 2 \\
3 & 4
\end{pmatrix}
\]
Saili le fa'afeagai o le matrix \( C \).
Talanoaga:
Mo se matrix 2×2, e mafai ona fuafuaina le inverse e faʻaaoga ai le fua faʻatatau:
\[
C^{-1} = \frac{1}{\text{det}(C)} \begin{pmatrix}
d & -b \\
-c ma le a
\end{pmatrix}
\]
lea \( C = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).
Muamua, matou te fuafuaina le mea e fuafua ai le matrix \( C \):
\[
\text{det}(C) = (1 \cdot 4) – (2 \cdot 3) = 4 – 6 = -2
\]
Ona sui lea i le fua fa'afeagai:
\[
C^{-1} = \frac{1}{-2} \begin{pmatrix}
4 ma le -2
-3 & 1
\end{pmatrix}
= \begin{pmatrix}
-2 ma le 1
\frac{3}{2} & -\frac{1}{2}
\end{pmatrix}
\]
O lea la, o le fa'afeagai o le matrix \( C \) o le \( \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \).
Fa'ata'ita'iga Fesili 4: Fa'afeagai o se Matrix 3×3
I le tuuina atu o le matrix \( D \) e pei ona taua i lalo:
\[
D = \begin{pmatrix}
2 & 0 & 1 \\
3 & 0 & 0 \\
1&4&2
\end{pmatrix}
\]
Saili le fa'afeagai o le matrix \( D \).
Talanoaga:
Mo matrices 3×3 po'o le n×n, o le metotia masani e fa'aaogaina o le metotia echelon po'o le metotia adjoint. O iinei, o le a tatou fa'aaogaina ai le metotia echelon.
O le laasaga muamua o le fausiaina lea o le augmented matrix \( [D|I] \) lea o le \( I \) o le identity matrix:
\[
\left[\begin{array}{ccc|ccc}
2 & 0 & 1 & 1 & 0 & 0 \\
3 & 0 & 0 & 0 & 1 & 0 \\
1 & 4 & 2 & 0 & 0 & 1
\end{array}\right]
\]
Ona, faatino lea o galuega faatino o laina faavae seia oo ina tatou fausia le matrix o le identity i le agavale:
1. Laina 1: \( B_1 \div 2 \)
\[
\left[\begin{array}{ccc|ccc}
1 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
3 & 0 & 0 & 0 & 1 & 0 \\
1 & 4 & 2 & 0 & 0 & 1
\end{array}\right]
\]
2. Laina 2: \( B_2 – 3B_1 \)
\[
\left[\begin{array}{ccc|ccc}
1 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & -\frac{3}{2} & -\frac{3}{2} & 1 & 0 \\
1 & 4 & 2 & 0 & 0 & 1
\end{array}\right]
\]
3. Laina 3: \( B_3 – B_1 \)
\[
\left[\begin{array}{ccc|ccc}
1 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & -\frac{3}{2} & -\frac{3}{2} & 1 & 0 \\
0 & 4 & \frac{3}{2} & -\frac{1}{2} & 0 & 1
\end{array}\right]
\]
4. Laina 3: \( B_3 \div 4 \)
\[
\left[\begin{array}{ccc|ccc}
1 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & -\frac{3}{2} & -\frac{3}{2} & 1 & 0 \\
0 & 1 & \frac{3}{8} & -\frac{1}{8} & 0 & \frac{1}{4}
\end{array}\right]
\]
5. Laina 1: \( B_1 – \frac{1}{2}B_3 \)
\[
\left[\begin{array}{ccc|ccc}
1 & 0 & 0 & \frac{5}{16} & 0 & -\frac{1}{8} \\
0 & 0 & -\frac{3}{2} & -\frac{3}{2} & 1 & 0 \\
0 & 1 & \frac{3}{8} & -\frac{1}{8} & 0 & \frac{1}{4}
\end{array}\right]
\]
6. Laina 2: \( B_2 \div -\frac{3}{2} \)
\[
\left[\begin{array}{ccc|ccc}
1 & 0 & 0 & \frac{5}{16} & 0 & -\frac{1}{8} \\
0 & 0 & 1 & 1 & -\frac{2}{3} & 0 \\
0 & 1 & \frac{3}{8} & -\frac{1}{8} & 0 & \frac{1}{4}
\end{array}\right]
\]
7. Laina 3: \( B_3 – \frac{3}{8} B_2 \)
\[
\left[\begin{array}{ccc|ccc}
1 & 0 & 0 & \frac{5}{16} & 0 & -\frac{1}{8} \\
0 & 0 & 1 & 1 & -\frac{2}{3} & 0 \\
0 & 1 & 0 & -\frac{1}{4} & \frac{1}{6} & \frac{1}{4}
\end{array}\right]
\]
O lea la, o le fa'afeagai o le matrix \( D \) o le \( \begin{pmatrix} \frac{5}{16} & 0 & -\frac{1}{8} \\ 1 & -\frac{2}{3} & 0 \\ -\frac{1}{4} & \frac{1}{6} & \frac{1}{4} \end{pmatrix} \).
Faatasi ai ma se malamalamaga i manatu ma faataitaiga patino, e mafai ona tatou iloa ai o le fuafuaina o mea e fuafua ma mea e faafeagai ma matrices e mafai ona faia e ala i le faaaogaina o metotia faigofie, ae e iai sona aafiaga tele i le auiliiliga o faamatalaga ma le foiaina o faafitauli faigata o le matematika. O lenei malamalamaga e taua tele i le tele o talosaga, e aofia ai ata komepiuta, auiliiliga o faamatalaga, ma faiga o fua faatatau laina.
