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MATLAB training -Matrix Determinant
๊ณตํ๋ฏธ์ ๋ถํII (์ฅํ์ ๊ต์๋) - ๋ฐ์ง์ ์์
์กฐ๊ต - Strike while the iron is hot. Practice makes perfect!
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Matrix Determinant A ๐ร๐ matrix
A linear system of 2 equations in 2 unknowns, ๐ฅ 1 , ๐ฅ 2 is a set of equations of the form Matrix form of the linear system Component form ๐= ๐จ โ๐ ๐ ? : Gauss elimination, Crammerโs rule, Matrix inversion, etc. ๐ 11 ๐ฅ 1 + ๐ 12 ๐ฅ 2 = ๐ 1 ๐ 21 ๐ฅ 1 + ๐ 22 ๐ฅ 2 = ๐ 2 ๐จ๐=๐ ๐= ๐ฅ 1 ๐ฅ 2 , ๐= ๐ 1 ๐ 2 ๐= ๐ 11 ๐ 12 ๐ 21 ๐ 22 , [๐ ๐๐ ] ๐ฅ ๐ = ๐ ๐
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Matrix Determinant Second order determinant A determinant of order 2
Example 1. Example 2.(Symmetric, ๐ด ๐ =๐ด) Example 3. (Skew-symmetric, ๐ด ๐ =โ๐ด) D=det ๐จ = ๐ 11 ๐ 12 ๐ 21 ๐ 22 = ๐ 11 ๐ 22 โ ๐ 12 ๐ 21 =4โ5โ2โ3=14 1 โ2 โ2 6 =1โ6โ โ2 โ โ2 =2 cos ๐๐ sin ๐๐ โ sin ๐๐ cos ๐๐ = cos ๐๐ โ cos ๐๐ โ sin ๐๐ โ โ sin ๐๐ =1
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Matrix Determinant Third order determinant A determinant of order 3
Example 1. Example 2. Example 3. (skew-symmetrix) D=det ๐จ = ๐ 11 ๐ 12 ๐ 13 ๐ 21 ๐ 22 ๐ 23 ๐ 31 ๐ 32 ๐ 33 = ๐=1 3 โ1 ๐+๐ ๐ ๐,๐ ๐ ๐,๐ , (for a fixed ๐, Minor ๐ ๐,๐ ) ๐ท= โ =1โ โ โ โ โ โ โ โ1 0 =โ12 ๐ท= โ โ =โ3โ โ =โ60 ๐ท= 0 3 โ1 โ3 0 โ =3โ โ โ3 โ โ1โ โ โ =0
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Matrix Determinant MATLAB A determinant of order 3
- The minor ๐ ๐,๐ is defined to be the determinant of the ๐โ1 ร ๐โ1 -matrix that results from ๐ด by removing the ๐ -th row and the ๐-th column. 3. Algorithm Define: a matrix ๐จ Define temporary ๐จ: At ๐=1 ๐ท=0 for ๐=1:3 At(๐, :) = []; At(:,๐) =[]; ๐ท=๐ท+ โ1 ๐+๐ ร๐ด ๐,๐ ร(At(2,2)โAt(1,1) - At(1,2) โ At(2,1)) At=๐จ; end D=det ๐จ = ๐ 11 ๐ 12 ๐ 13 ๐ 21 ๐ 22 ๐ 23 ๐ 31 ๐ 32 ๐ 33 = ๐=1 3 โ1 ๐+๐ ๐ ๐,๐ ๐ ๐,๐ = โ ๐ 1,1 ๐ 1,1 + โ ๐ 1,2 ๐ 1,2 + โ ๐ 1,3 ๐ 1,3 ,
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Matrix Determinant MATLAB Inverse of a Matrix
where ๐ถ ๐๐ is the cofactor of ๐ ๐๐ in det ๐จ . Example 1. Example 2. ๐ด โ1 = 1 det ๐จ ๐ถ ๐๐ T = 1 det ๐จ ๐ถ 11 โฏ ๐ถ ๐1 โฎ โฑ โฎ ๐ถ 1๐ โฏ ๐ถ ๐๐ ๐จ= ๏ ๐จ โ1 = โ1 โ2 3 = 0.4 โ0.1 โ ๐จ= โ โ1 1 โ = 1 det ๐จ ๐ถ 11 ๐ถ 21 ๐ถ 31 ๐ถ 12 ๐ถ 22 ๐ถ 32 ๐ถ 13 ๐ถ 23 ๐ถ where ๐ถ ๐, ๐ : ๐ถ 11 = โ โ =โ7, ๐ถ 21 = โ =2, ๐ถ 31 = โ โ1 1 =3, ๐ถ 23 = โ โ1 1 โ1 3 =โ2, ๐ถ 12 = โ โ1 4 =โ13, ๐ถ 22 = โ โ1 2 โ1 4 =โ2, โฏ
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Matrix Determinant MATLAB Algorithm Define: a matrix ๐จ
Define temporary ๐จ: At ๐=1 Initial determinant: ๐ท=0 for ๐=1:3 At(๐, :) = []; At(:,๐) =[]; ๐ท=๐ท+ โ1 ๐+๐ ร๐ด ๐,๐ ร(At(2,2)โAt(1,1) - At(1,2) โ At(2,1)) At=๐จ; end Temporary A : CF=๐จ for ๐=1:3 CF ๐,: = []; CF(:,๐) =[]; Inv_A ๐,๐ = โ1 ๐+๐ CF 2,2 CF 1,1 โCF(1,2)CF(2,1) CF=A; Inverse of a matrix A: Inv_A=Inv_A/DET
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๏ โLECTUREโ ๏ Undergraduate (๊ณตํ๋ฏธ์ ๋ถํII)
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