Ajụjụ Ihe Nlereanya Na-atụle Ihe Ndị A Ga-ekpebi na Ihe Ndị A Ga-agbanwe na Matrix Inversions
Ihe ndị na-ekpebi matriks na ihe ndị na-agbanwe agbanwe matriks bụ echiche abụọ dị mkpa na algebra linear nke nwere ọtụtụ ojiji n'ọtụtụ ngalaba, gụnyere mgbakọ na mwepụ, fisiksi, akụnụba, na injinia. Nghọta zuru oke nke echiche ndị a dị oke mkpa maka idozi ọtụtụ nsogbu mgbakọ na mwepụ dị mgbagwoju anya. N'isiokwu a, anyị ga-atụle ihe atụ nke ihe ndị na-ekpebi matriks na ihe ndị na-agbanwe agbanwe, yana mkparịta ụka zuru oke.
Ihe Nchọpụta Matrix
Ihe na-ekpebi ihe bụ scalar nke ejikọtara ya na matriks sụkwịa (matriks nwere otu ọnụọgụ ahịrị na kọlụm). Ihe na-ekpebi ihe nwere ike inye ozi dị mkpa gbasara ihe onwunwe nke matriks ahụ, dịka ma ọ bụ ihe a na-apụghị ịgbanwe agbanwe ma ọ bụ na ọ bụghị.
Ajụjụ Ihe Nlereanya nke 1: Ihe Nchọpụta nke Matrix 2×2
E nyere matriks \( A \) dị ka ndị a:
\[
A = \malite{pmatrix}
4 na 3 \\
2 & 1
\ọgwụgwụ{pmatrix}
\]
Chọpụta ihe na-ekpebi matriks \( A \).
Azịza:
Maka matrix 2×2, enwere ike ịgbakọ ihe na-ekpebi ihe site na iji usoro dị mfe a:
\[
\text{det}(A) = mgbasa ozi – bc
\]
ebe \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).
Mgbanwe nke ihe dị na matriks \( A \):
\[
\text{det}(A) = (4 \ugboro 1) – (3 \ugboro 2) = 4 - 6 = -2
\]
Ya mere, ihe na-ekpebi matriks \(A \) bụ -2.
Ajụjụ Ihe Nlereanya nke 2: Ihe Nchọpụta nke Matrix 3×3
E nyere matriks \( B \) dị ka ndị a:
\[
B = \malite{pmatrix}
1 na 2 na 3 \\
0 na 1 na 4 \\
5 & 6 & 0
\ọgwụgwụ{pmatrix}
\]
Chọpụta ihe na-ekpebi matriks \( B \).
Azịza:
Maka matrix 3×3, enwere ike ịgbakọ ihe na-ekpebi ihe site na iji iwu Sarrus ma ọ bụ cofactors. N'ebe a, anyị ga-eji iwu Sarrus mee ka mgbakọ ahụ dị mfe.
Mee ka ogidi abụọ mbụ dị n'akụkụ aka nri nke matriks ahụ dị ka nke a:
\[
\text{det}(B) = malite{vmatrix}
1 na 2 na 3 \\
0 na 1 na 4 \\
5 & 6 & 0
\ọgwụgwụ{vmatrix}
= 1\cdot1\cdot0 + 2\cdot4\cdot5 + 3\cdot0\cdot6 – (3\cdot1\cdot5 + 2\cdot0\cdot0 + 1\cdot4\cdot6)
\]
\[
= 0 + 40 + 0 – (15 + 0 + 24)
\]
\[
= 40 - 39 = 1
\]
Ya mere, ihe na-ekpebi matriks \(B \) bụ 1.
Matriks Mgbanwe
Mgbanwe nke matriks \(A \) (ọ bụrụ na ọ dị) bụ matriks \(A^{-1} \) nke na-emezu ọnọdụ ndị a:
\[
A \cdot A^{-1} = A^{-1} \cdot A = I
\]
ebe \(I \) bụ matriks njirimara nke ihe diagonal ya bụ 1 na ihe ndị ọzọ bụ 0.
Ihe atụ Ajụjụ nke 3: Mgbanwe nke Matrix 2×2
E nyere matriks \(C \) dị ka ndị a:
\[
C = mmalite{pmatrix}
1 na 2 \\
3 & 4
\ọgwụgwụ{pmatrix}
\]
Chọta mgbanwe nke matriks \(C \).
Azịza:
Maka matriks 2×2, enwere ike ịgbakọ inverse site na iji usoro a:
\[
C^{-1} = \frac{1}{\text{det}(C)} \begin{pmatrix}
d na -b \\
-c na a
\ọgwụgwụ{pmatrix}
\]
ebe \(C = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).
Nke mbụ, anyị na-agbakọ ihe na-ekpebi matriks \(C \):
\[
\ederede{det}(C) = (1 \cdot 4) - (2 \ cdot 3) = 4 - 6 = -2
\]
Mgbe ahụ, tinye ya n'ime usoro ntụgharị:
\[
C^{-1} = \frac{1}{-2} \begin{pmatrix}
4 na -2 \\
-3 & 1
\ọgwụgwụ{pmatrix}
= \malite{pmatrix}
-2 na 1 \\
\frac{3}{2} na -\frac{1}{2}
\ọgwụgwụ{pmatrix}
\]
Ya mere, ihe dị iche na matrix \(C \) bụ \( \begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \).
Ihe atụ Ajụjụ nke 4: Mgbanwe nke Matrix 3×3
E nyere matriks \( D \) dị ka ndị a:
\[
D = \malite{pmatrix}
2 na 0 na 1 \\
3 na 0 na 0 \\
1 & 4 & 2
\ọgwụgwụ{pmatrix}
\]
Chọta ihe dị iche na matriks \( D \).
Azịza:
Maka matrices 3×3 maọbụ n×n, usoro a na-ejikarị eme ihe bụ usoro echelon maọbụ usoro adjoint. N'ebe a, anyị ga-eji usoro echelon.
Nzọụkwụ mbụ bụ ịmepụta matrix agbakwunyere \( [D|I] \) ebe \(I \) bụ matrix njirimara:
\[
\left[\begin{array}{ccc|ccc}
2 na 0 na 1 na 1 na 0 na 0 \\
3 na 0 na 0 na 0 na 1 na 0 \\
1 na 4 na 2 na 0 na 0 na 1
\ọgwụgwụ{array}\nri]
\]
Mgbe ahụ, rụọ ọrụ ahịrị mbụ ruo mgbe anyị mepụtara matrix njirimara n'aka ekpe:
1. Ahịrị nke 1: \( B_1 \div 2 \)
\[
\left[\begin{array}{ccc|ccc}
1 na 0 na \frac{1}{2} na \frac{1}{2} na 0 na 0 \\
3 na 0 na 0 na 0 na 1 na 0 \\
1 na 4 na 2 na 0 na 0 na 1
\ọgwụgwụ{array}\nri]
\]
2. Ahịrị nke 2: \( B_2 – 3B_1 \)
\[
\left[\begin{array}{ccc|ccc}
1 na 0 na \frac{1}{2} na \frac{1}{2} na 0 na 0 \\
0 na 0 na -\frac{3}{2} na -\frac{3}{2} na 1 na 0 \\
1 na 4 na 2 na 0 na 0 na 1
\ọgwụgwụ{array}\nri]
\]
3. Ahịrị nke 3: \( B_3 – B_1 \)
\[
\left[\begin{array}{ccc|ccc}
1 na 0 na \frac{1}{2} na \frac{1}{2} na 0 na 0 \\
0 na 0 na -\frac{3}{2} na -\frac{3}{2} na 1 na 0 \\
0 na 4 na \frac{3}{2} na -\frac{1}{2} na 0 na 1
\ọgwụgwụ{array}\nri]
\]
4. Ahịrị nke 3: \( B_3 \div 4 \)
\[
\left[\begin{array}{ccc|ccc}
1 na 0 na \frac{1}{2} na \frac{1}{2} na 0 na 0 \\
0 na 0 na -\frac{3}{2} na -\frac{3}{2} na 1 na 0 \\
0 na 1 na \frac{3}{8} na -\frac{1}{8} na 0 na \frac{1}{4}
\ọgwụgwụ{array}\nri]
\]
5. Ahịrị nke 1: \( B_1 – \frac{1}{2}B_3 \)
\[
\left[\begin{array}{ccc|ccc}
1 na 0 na 0 na \frac{5}{16} na 0 na -\frac{1}{8} \\
0 na 0 na -\frac{3}{2} na -\frac{3}{2} na 1 na 0 \\
0 na 1 na \frac{3}{8} na -\frac{1}{8} na 0 na \frac{1}{4}
\ọgwụgwụ{array}\nri]
\]
6. Ahịrị nke 2: \( B_2 \div -\frac{3}{2} \)
\[
\left[\begin{array}{ccc|ccc}
1 na 0 na 0 na \frac{5}{16} na 0 na -\frac{1}{8} \\
0 na 0 na 1 na 1 na -\frac{2}{3} na 0 \\
0 na 1 na \frac{3}{8} na -\frac{1}{8} na 0 na \frac{1}{4}
\ọgwụgwụ{array}\nri]
\]
7. Ahịrị nke 3: \( B_3 – \frac{3}{8} B_2 \)
\[
\left[\begin{array}{ccc|ccc}
1 na 0 na 0 na \frac{5}{16} na 0 na -\frac{1}{8} \\
0 na 0 na 1 na 1 na -\frac{2}{3} na 0 \\
0 na 1 na 0 na -\frac{1}{4} na \frac{1}{6} na \frac{1}{4}
\ọgwụgwụ{array}\nri]
\]
Ya mere, ihe dị iche na matrix \( D \) bụ \( \begin{pmatrix} \frac{5}{16} & 0 & -\frac{1}{8} \\ 1 & -\frac{2}{3} & 0 \\ -\frac{1}{4} & \frac{1}{6} & \frac{1}{4} \end{pmatrix} \).
Site n'ịghọta echiche na ihe atụ ndị doro anya, anyị nwere ike ịhụ na enwere ike iji ụzọ dị mfe mee ịgbakọ ihe ndị na-ekpebi matriks na mgbanwe, mana ọ na-enwe mmetụta dị ukwuu na nyocha data na idozi nsogbu mgbakọ na mwepụ ndị siri ike karị. Nghọta a dị oke mkpa n'ọtụtụ ngwa, gụnyere eserese kọmputa, nyocha data, na sistemụ nha nhata ahịrị.
