1. MATLAB training -Matrix Determinant
공학미적분학II (장택수 교수님)
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2. Matrix Determinant A 𝟐×𝟐 matrix
A linear system of 2 equations in 2 unknowns, 𝑥 1 , 𝑥 2 is a set of equations of the form
Matrix form of the linear system
Component form
𝒙= 𝑨 −𝟏 𝒃 ? : Gauss elimination, Crammer’s rule, Matrix inversion, etc.
𝑎 11 𝑥 1 + 𝑎 12 𝑥 2 = 𝑏 1
𝑎 21 𝑥 1 + 𝑎 22 𝑥 2 = 𝑏 2
𝑨𝒙=𝒃
𝒙= 𝑥 1 𝑥 2 ,
𝒃= 𝑏 1 𝑏 2
𝐀= 𝑎 11 𝑎 12 𝑎 21 𝑎 22 ,
[𝑎 𝑖𝑗 ] 𝑥 𝑗 = 𝑏 𝑖

3. Matrix Determinant Second order determinant A determinant of order 2
Example 1.
Example 2.(Symmetric, 𝐴 𝑇 =𝐴)
Example 3. (Skew-symmetric, 𝐴 𝑇 =−𝐴)
D=det 𝑨 = 𝑎 11 𝑎 12 𝑎 21 𝑎 22 = 𝑎 11 𝑎 22 − 𝑎 12 𝑎 21
=4∙5−2∙3=14
1 −2 −2 6 =1∙6− −2 ∙ −2 =2
cos 𝑛𝜃 sin 𝑛𝜃 − sin 𝑛𝜃 cos 𝑛𝜃 = cos 𝑛𝜃 ∙ cos 𝑛𝜃 − sin 𝑛𝜃 ∙ − sin 𝑛𝜃 =1

4. Matrix Determinant Third order determinant A determinant of order 3
Example 1.
Example 2.
Example 3. (skew-symmetrix)
D=det 𝑨 = 𝑎 11 𝑎 12 𝑎 13 𝑎 21 𝑎 22 𝑎 23 𝑎 31 𝑎 32 𝑎 33 = 𝑗=1 3 −1 𝑖+𝑗 𝑎 𝑖,𝑗 𝑀 𝑖,𝑗 , (for a fixed 𝑖, Minor 𝑀 𝑖,𝑗 )
𝐷= − =1∙ − ∙ − − ∙ − −1 0 =−12
𝐷= − − =−3∙ − =−60
𝐷= 0 3 −1 −3 0 − =3∙ − −3 − −1∙ − − =0

5. Matrix Determinant MATLAB A determinant of order 3
- The minor 𝑀 𝑖,𝑗 is defined to be the determinant of the 𝑛−1 × 𝑛−1 -matrix that results from 𝐴 by removing the 𝑖 -th row and the 𝑗-th column.
3. Algorithm
Define: a matrix 𝑨
Define temporary 𝑨: At
𝑖=1
𝐷=0
for 𝑗=1:3
At(𝑖, :) = [];
At(:,𝑗) =[];
𝐷=𝐷+ −1 𝑖+𝑗 ×𝐴 𝑖,𝑗 ×(At(2,2)∙At(1,1) - At(1,2) ∙ At(2,1))
At=𝑨;
end
D=det 𝑨 = 𝑎 11 𝑎 12 𝑎 13 𝑎 21 𝑎 22 𝑎 23 𝑎 31 𝑎 32 𝑎 33 = 𝑗=1 3 −1 𝑖+𝑗 𝑎 𝑖,𝑗 𝑀 𝑖,𝑗
= − 𝑎 1,1 𝑀 1,1 + − 𝑎 1,2 𝑀 1,2 + − 𝑎 1,3 𝑀 1,3 ,

6. Matrix Determinant MATLAB Inverse of a Matrix
where 𝐶 𝑗𝑘 is the cofactor of 𝑎 𝑗𝑘 in det 𝑨 .
Example 1.
Example 2.
𝐴 −1 = 1 det 𝑨 𝐶 𝑗𝑘 T = 1 det 𝑨 𝐶 11 ⋯ 𝐶 𝑛1 ⋮ ⋱ ⋮ 𝐶 1𝑛 ⋯ 𝐶 𝑛𝑛
𝑨=  𝑨 −1 = −1 −2 3 = 0.4 −0.1 −
𝑨= − −1 1 − = 1 det 𝑨 𝐶 11 𝐶 21 𝐶 31 𝐶 12 𝐶 22 𝐶 32 𝐶 13 𝐶 23 𝐶 where 𝐶 𝑖, 𝑗 :
𝐶 11 = − − =−7,
𝐶 21 = − =2,
𝐶 31 = − −1 1 =3,
𝐶 23 = − −1 1 −1 3 =−2,
𝐶 12 = − −1 4 =−13,
𝐶 22 = − −1 2 −1 4 =−2, ⋯

7. Matrix Determinant MATLAB Algorithm Define: a matrix 𝑨
Define temporary 𝑨: At
𝑖=1
Initial determinant: 𝐷=0
for 𝑗=1:3
At(𝑖, :) = [];
At(:,𝑗) =[];
𝐷=𝐷+ −1 𝑖+𝑗 ×𝐴 𝑖,𝑗 ×(At(2,2)∙At(1,1) - At(1,2) ∙ At(2,1))
At=𝑨;
end
Temporary A : CF=𝑨
for 𝑘=1:3
CF 𝑗,: = [];
CF(:,𝑘) =[];
Inv_A 𝑘,𝑗 = −1 𝑗+𝑘 CF 2,2 CF 1,1 −CF(1,2)CF(2,1)
CF=A;
Inverse of a matrix A: Inv_A=Inv_A/DET

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