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MATLAB training -Matrix Determinant

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1 MATLAB training -Matrix Determinant
๊ณตํ•™๋ฏธ์ ๋ถ„ํ•™II (์žฅํƒ์ˆ˜ ๊ต์ˆ˜๋‹˜) - ๋ฐ•์ง„์ˆ˜ ์ˆ˜์—… ์กฐ๊ต - Strike while the iron is hot. Practice makes perfect!

2 Matrix Determinant A ๐Ÿร—๐Ÿ matrix
A linear system of 2 equations in 2 unknowns, ๐‘ฅ 1 , ๐‘ฅ 2 is a set of equations of the form Matrix form of the linear system Component form ๐’™= ๐‘จ โˆ’๐Ÿ ๐’ƒ ? : Gauss elimination, Crammerโ€™s rule, Matrix inversion, etc. ๐‘Ž 11 ๐‘ฅ 1 + ๐‘Ž 12 ๐‘ฅ 2 = ๐‘ 1 ๐‘Ž 21 ๐‘ฅ 1 + ๐‘Ž 22 ๐‘ฅ 2 = ๐‘ 2 ๐‘จ๐’™=๐’ƒ ๐’™= ๐‘ฅ 1 ๐‘ฅ 2 , ๐’ƒ= ๐‘ 1 ๐‘ 2 ๐€= ๐‘Ž 11 ๐‘Ž 12 ๐‘Ž 21 ๐‘Ž 22 , [๐‘Ž ๐‘–๐‘— ] ๐‘ฅ ๐‘— = ๐‘ ๐‘–

3 Matrix Determinant Second order determinant A determinant of order 2
Example 1. Example 2.(Symmetric, ๐ด ๐‘‡ =๐ด) Example 3. (Skew-symmetric, ๐ด ๐‘‡ =โˆ’๐ด) D=det ๐‘จ = ๐‘Ž 11 ๐‘Ž 12 ๐‘Ž 21 ๐‘Ž 22 = ๐‘Ž 11 ๐‘Ž 22 โˆ’ ๐‘Ž 12 ๐‘Ž 21 =4โˆ™5โˆ’2โˆ™3=14 1 โˆ’2 โˆ’2 6 =1โˆ™6โˆ’ โˆ’2 โˆ™ โˆ’2 =2 cos ๐‘›๐œƒ sin ๐‘›๐œƒ โˆ’ sin ๐‘›๐œƒ cos ๐‘›๐œƒ = cos ๐‘›๐œƒ โˆ™ cos ๐‘›๐œƒ โˆ’ sin ๐‘›๐œƒ โˆ™ โˆ’ sin ๐‘›๐œƒ =1

4 Matrix Determinant Third order determinant A determinant of order 3
Example 1. Example 2. Example 3. (skew-symmetrix) D=det ๐‘จ = ๐‘Ž 11 ๐‘Ž 12 ๐‘Ž 13 ๐‘Ž 21 ๐‘Ž 22 ๐‘Ž 23 ๐‘Ž 31 ๐‘Ž 32 ๐‘Ž 33 = ๐‘—=1 3 โˆ’1 ๐‘–+๐‘— ๐‘Ž ๐‘–,๐‘— ๐‘€ ๐‘–,๐‘— , (for a fixed ๐‘–, Minor ๐‘€ ๐‘–,๐‘— ) ๐ท= โˆ’ =1โˆ™ โˆ’ โˆ™ โˆ’ โˆ’ โˆ™ โˆ’ โˆ’1 0 =โˆ’12 ๐ท= โˆ’ โˆ’ =โˆ’3โˆ™ โˆ’ =โˆ’60 ๐ท= 0 3 โˆ’1 โˆ’3 0 โˆ’ =3โˆ™ โˆ’ โˆ’3 โˆ’ โˆ’1โˆ™ โˆ’ โˆ’ =0

5 Matrix Determinant MATLAB A determinant of order 3
- The minor ๐‘€ ๐‘–,๐‘— is defined to be the determinant of the ๐‘›โˆ’1 ร— ๐‘›โˆ’1 -matrix that results from ๐ด by removing the ๐‘– -th row and the ๐‘—-th column. 3. Algorithm Define: a matrix ๐‘จ Define temporary ๐‘จ: At ๐‘–=1 ๐ท=0 for ๐‘—=1:3 At(๐‘–, :) = []; At(:,๐‘—) =[]; ๐ท=๐ท+ โˆ’1 ๐‘–+๐‘— ร—๐ด ๐‘–,๐‘— ร—(At(2,2)โˆ™At(1,1) - At(1,2) โˆ™ At(2,1)) At=๐‘จ; end D=det ๐‘จ = ๐‘Ž 11 ๐‘Ž 12 ๐‘Ž 13 ๐‘Ž 21 ๐‘Ž 22 ๐‘Ž 23 ๐‘Ž 31 ๐‘Ž 32 ๐‘Ž 33 = ๐‘—=1 3 โˆ’1 ๐‘–+๐‘— ๐‘Ž ๐‘–,๐‘— ๐‘€ ๐‘–,๐‘— = โˆ’ ๐‘Ž 1,1 ๐‘€ 1,1 + โˆ’ ๐‘Ž 1,2 ๐‘€ 1,2 + โˆ’ ๐‘Ž 1,3 ๐‘€ 1,3 ,

6 Matrix Determinant MATLAB Inverse of a Matrix
where ๐ถ ๐‘—๐‘˜ is the cofactor of ๐‘Ž ๐‘—๐‘˜ in det ๐‘จ . Example 1. Example 2. ๐ด โˆ’1 = 1 det ๐‘จ ๐ถ ๐‘—๐‘˜ T = 1 det ๐‘จ ๐ถ 11 โ‹ฏ ๐ถ ๐‘›1 โ‹ฎ โ‹ฑ โ‹ฎ ๐ถ 1๐‘› โ‹ฏ ๐ถ ๐‘›๐‘› ๐‘จ= ๏ƒ  ๐‘จ โˆ’1 = โˆ’1 โˆ’2 3 = 0.4 โˆ’0.1 โˆ’ ๐‘จ= โˆ’ โˆ’1 1 โˆ’ = 1 det ๐‘จ ๐ถ 11 ๐ถ 21 ๐ถ 31 ๐ถ 12 ๐ถ 22 ๐ถ 32 ๐ถ 13 ๐ถ 23 ๐ถ where ๐ถ ๐‘–, ๐‘— : ๐ถ 11 = โˆ’ โˆ’ =โˆ’7, ๐ถ 21 = โˆ’ =2, ๐ถ 31 = โˆ’ โˆ’1 1 =3, ๐ถ 23 = โˆ’ โˆ’1 1 โˆ’1 3 =โˆ’2, ๐ถ 12 = โˆ’ โˆ’1 4 =โˆ’13, ๐ถ 22 = โˆ’ โˆ’1 2 โˆ’1 4 =โˆ’2, โ‹ฏ

7 Matrix Determinant MATLAB Algorithm Define: a matrix ๐‘จ
Define temporary ๐‘จ: At ๐‘–=1 Initial determinant: ๐ท=0 for ๐‘—=1:3 At(๐‘–, :) = []; At(:,๐‘—) =[]; ๐ท=๐ท+ โˆ’1 ๐‘–+๐‘— ร—๐ด ๐‘–,๐‘— ร—(At(2,2)โˆ™At(1,1) - At(1,2) โˆ™ At(2,1)) At=๐‘จ; end Temporary A : CF=๐‘จ for ๐‘˜=1:3 CF ๐‘—,: = []; CF(:,๐‘˜) =[]; Inv_A ๐‘˜,๐‘— = โˆ’1 ๐‘—+๐‘˜ CF 2,2 CF 1,1 โˆ’CF(1,2)CF(2,1) CF=A; Inverse of a matrix A: Inv_A=Inv_A/DET

8 ๊ฐ•์˜ ์ž๋ฃŒ ๋ฐ ๋ฐœํ‘œ ์ž๋ฃŒ ์—…๋กœ๋“œ Where to? Homepage: oceanwave.pusan.ac.kr
๏ƒ  โ€œLECTUREโ€ ๏ƒ  Undergraduate (๊ณตํ•™๋ฏธ์ ๋ถ„ํ•™II)

9 THANK YOU


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