Eisimpleirean de Cheistean a’ Deasbad Determinants agus Maitrís Inbhirean
’S e dà bhun-bheachd ann an ailseabra loidhneach a th’ ann an dearcan maitrís agus inbhirean maitrís aig a bheil cleachdaidhean farsaing ann an diofar raointean, nam measg matamataig, fiosaig, eaconamas agus innleadaireachd. Tha tuigse mhionaideach air na bun-bheachdan seo riatanach airson mòran dhuilgheadasan matamataigeach iom-fhillte fhuasgladh. San artaigil seo, bruidhnidh sinn air eisimpleirean de dh’earcan maitrís agus inbhirean, còmhla ri deasbad coileanta.
Dearbhadh Matrix
’S e sgalar a th’ anns an determinant a tha co-cheangailte ri maitrís ceàrnagach (maitrís leis an aon àireamh de shreathan is cholbhan). Faodaidh an determinant fiosrachadh cudromach a thoirt seachad mu fheartan a’ mhaitrís, leithid a bheil e neo-thionndaidheach no nach eil.
Eisimpleir Ceist 1: Dearbhadh Maitrís 2 × 2
Air a thoirt seachad leis a’ mhaitrice \(A \) mar a leanas:
\[
A = \begin{pmatrix}
4 & 3 \\
2 & 1
\end{pmatrix}
\]
Obraich a-mach determinant a’ mhaitris \(A \).
Deasbad:
Airson maitrís 2 × 2, faodar an determinant obrachadh a-mach leis an fhoirmle shìmplidh a leanas:
\[
\text{det}(A) = ad – bc
\]
far a bheil (A = \begin{pmatrix} a & b \\c & d \end{pmatrix}).
Ionadachadh eileamaidean a’ mhaitris \(A \):
\[
\text{det}(A) = (4 × 1) – (3 × 2) = 4 – 6 = -2
\]
Mar sin, is e -2 determinant a’ mhaitris \(A \).
Eisimpleir Ceist 2: Dearbhadh Maitrís 3 × 3
Air a thoirt seachad leis a’ mhaitrice \(B \) mar a leanas:
\[
B = \begin{pmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{pmatrix}
\]
Obraich a-mach determinant a’ mhaitris \(B \).
Deasbad:
Airson maitrís 3 × 3, faodar an determinant obrachadh a-mach le bhith a’ cleachdadh riaghailt Sarrus no co-fhactaran. An seo, cleachdaidh sinn riaghailt Sarrus gus an àireamhachadh a dhèanamh nas sìmplidhe.
Dèan lethbhreac den chiad dà cholbh air taobh deas a’ mhaitris:
\[
\text{det}(B) = \tòisich{vmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
5 & 6 & 0
\end{vmatrix}
= 1\cdot1\cdot0 + 2\cdot4\cdot5 + 3\cdot0\cdot6 – (3\cdot1\cdot5 + 2\cdot0\cdot0 + 1\cdot4\cdot6)
\]
\[
= 0 + 40 + 0 – (15 + 0 + 24)
\]
\[
= 40 – 39 = 1
\]
Mar sin, is e 1 determinant a’ mhaitrice \(B \).
Maitrís neo-dhìreach
Is e maitrís A (-1) a choinnicheas ris na cumhaichean a leanas an taobh eile de mhaitrís \(A \) (ma tha e ann):
\[
A −1 = A −1 A = I
\]
far a bheil \(I \) na mhaitris dearbh-aithne aig a bheil na h-eileamaidean trastanach 1 agus na h-eileamaidean eile 0.
Eisimpleir Ceist 3: Inbhears Maitrís 2 × 2
Air a thoirt seachad leis a’ mhaitrice \(C \) mar a leanas:
\[
C = \begin{pmatrix}
1 & 2 \\
3 & 4
\end{pmatrix}
\]
Lorg an taobh eile den mhaitris \(C \).
Deasbad:
Airson maitrís 2 × 2, faodar an taobh eile obrachadh a-mach leis an fhoirmle:
\[
C^{-1} = \frac{1}{\text{det}(C)} \begin{pmatrix}
d & -b \\
-c & a
\end{pmatrix}
\]
far a bheil (C = (begin{pmatrix} a & b \\ c & d \end{pmatrix})).
An toiseach, obraichidh sinn a-mach determinant a’ mhaitris \( C \):
\[
\text{det}(C) = (1 \cdot 4) – (2 \cdot 3) = 4 – 6 = -2
\]
An uairsin, cuir an àite an fhoirmle neo-dhìreach:
\[
C^{-1} = \frac{1}{-2} \begin{pmatrix}
4 & -2 \\
-3 & 1
\end{pmatrix}
= \begin{pmatrix}
-2 & 1
\frac{3}{2} & -\frac{1}{2}
\end{pmatrix}
\]
Mar sin, is e bun-os-cionn a’ mhaitrice C (\begin{pmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{pmatrix} \).
Eisimpleir Ceist 4: Inbhears Maitrís 3 × 3
Air a thoirt seachad leis a’ mhaitrice \(D \) mar a leanas:
\[
D = \begin{pmatrix}
2 & 0 & 1 \\
3 & 0 & 0 \\
1 & 4 & 2
\end{pmatrix}
\]
Lorg an taobh eile den mhaitris \(D \).
Deasbad:
Airson maitrísean 3 × 3 no n × n, is e an dòigh echelon no an dòigh adjoint an dòigh chumanta a thathas a’ cleachdadh. An seo, cleachdaidh sinn an dòigh echelon.
Is e a’ chiad cheum am maitrís leasaichte a chruthachadh \( [D|I] \) far a bheil \( I \) na mhaitrís dearbh-aithne:
\[
\clì[\begin{array}{ccc|ccc}
2 & 0 & 1 & 1 & 0 & 0 \\
3 & 0 & 0 & 0 & 1 & 0 \\
1 & 4 & 2 & 0 & 0 & 1
\end{array}\right]
\]
An uairsin, dèan obrachaidhean sreath bunaiteach gus an cruthaich sinn am maitrís dearbh-aithne air an taobh chlì:
1. Loidhne 1: \( B_1 \div 2 \)
\[
\clì[\begin{array}{ccc|ccc}
1 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
3 & 0 & 0 & 0 & 1 & 0 \\
1 & 4 & 2 & 0 & 0 & 1
\end{array}\right]
\]
2. Sreath 2: \( B_2 – 3B_1 \)
\[
\clì[\begin{array}{ccc|ccc}
1 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & -\frac{3}{2} & -\frac{3}{2} & 1 & 0 \\
1 & 4 & 2 & 0 & 0 & 1
\end{array}\right]
\]
3. Loidhne 3: \( B_3 – B_1 \)
\[
\clì[\begin{array}{ccc|ccc}
1 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & -\frac{3}{2} & -\frac{3}{2} & 1 & 0 \\
0 & 4 & \frac{3}{2} & -\frac{1}{2} & 0 & 1
\end{array}\right]
\]
4. Loidhne 3: \( B_3 \div 4 \)
\[
\clì[\begin{array}{ccc|ccc}
1 & 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\
0 & 0 & -\frac{3}{2} & -\frac{3}{2} & 1 & 0 \\
0 & 1 & \frac{3}{8} & -\frac{1}{8} & 0 & \frac{1}{4}
\end{array}\right]
\]
5. Loidhne 1: \( B_1 – \frac{1}{2}B_3 \)
\[
\clì[\begin{array}{ccc|ccc}
1 & 0 & 0 & \frac{5}{16} & 0 & -\frac{1}{8} \\
0 & 0 & -\frac{3}{2} & -\frac{3}{2} & 1 & 0 \\
0 & 1 & \frac{3}{8} & -\frac{1}{8} & 0 & \frac{1}{4}
\end{array}\right]
\]
6. Loidhne 2: \( B_2 \div - \frac{3}{2} \)
\[
\clì[\begin{array}{ccc|ccc}
1 & 0 & 0 & \frac{5}{16} & 0 & -\frac{1}{8} \\
0 & 0 & 1 & 1 & -\frac{2}{3} & 0 \\
0 & 1 & \frac{3}{8} & -\frac{1}{8} & 0 & \frac{1}{4}
\end{array}\right]
\]
7. Loidhne 3: \( B_3 – \frac{3}{8} B_2 \)
\[
\clì[\begin{array}{ccc|ccc}
1 & 0 & 0 & \frac{5}{16} & 0 & -\frac{1}{8} \\
0 & 0 & 1 & 1 & -\frac{2}{3} & 0 \\
0 & 1 & 0 & -\frac{1}{4} & \frac{1}{6} & \frac{1}{4}
\end{array}\right]
\]
Mar sin, is e inbhir a’ mhaitris D (\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} \).
Le tuigse air na bun-bheachdan agus eisimpleirean concrait, chì sinn gum faodar obrachadh a-mach determinants agus inverses maitrísean le bhith a’ cleachdadh dhòighean caran sìmplidh, ach aig an aon àm a’ toirt buaidh mhòr air mion-sgrùdadh dàta agus fuasgladh dhuilgheadasan matamataigeach nas iom-fhillte. Tha an tuigse seo riatanach ann an grunn thagraidhean, a’ gabhail a-steach grafaigean coimpiutair, mion-sgrùdadh dàta, agus siostaman cho-aontaran loidhneach.
