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Линейная алгебра. MATLAB. МОДУЛЬ 1 занятие 2 определители и формулы крамера
Кучеренко Антон МП-18
20.09.2011
Упражнение 1
Вычисление определителей первого порядка
>> syms a11 a12 a21 a22
>> A=[a11 a12; a21 a22]
A =
[ a11, a12]
[ a21, a22]
>> detA=A(1,1)*A(2,2)-A(2,1)*A(1,2)
detA =
a11*a22-a21*a12
>> detA=det(A)
detA =
a11*a22-a21*a12
Упражнение 2
Вычислить определители второго порядка
>> A=[-1 4;-5 2]
A =
-1 4
-5 2
>> det(A)
ans =
18
>> syms a b
>> A=[a+b a-b;a+b a-b]
A =
[a+b, a-b]
[a+b, a-b]
>> det(A)
ans =
0
>> clear syms
>> syms x
>> A=[x x+1;-4 x+1]
A =
[ x, x+1]
[ -4, x+1]
>> det(A)
ans =
x^2+5*x+4
>> A=A(1,1)*A(2,2)-A(2,1)*A(1,2)
A =
x*(x+1)+4*x+4
Упражнение 3
3x-5y=13
2x+7y=81
>> A=[3 -5;2 7]
A =
3 -5
2 7
>> d=det(A)
d =
31
>> A1=[13 -5;81 7]
A1 =
13 -5
81 7
>> d1=det(A1)
d1 =
496
>> A2=[3 13; 2 81]
A2 =
3 13
2 81
>> d2=det(A2)
d2 =
217
>> x1=d1/d
x1 =
16
>> x2=d2/d
x2 =
7
>> clear A A1 A2 d d1 d2 x1 x2
3x-4y=-6
3x+4y=18
>> A=[3 -4;3 4]
A =
3 -4
3 4
>> d=det(A)
d =
24
>> A1=[3 -6;3 18]
A1 =
3 -6
3 18
>> d1=det(A1)
d1 =
72
>> A2=[-6 -4; 18 4]
A2 =
-6 -4
18 4
>> d2=det(A2)
d2 =
48
>> x1=d1/d
x1 =
3
>> x2=d2/d
x2 =
2
>> clear all
>> syms a1 a2 a3 b1 b2 b3 c1 c2 c3
Упражнение 4
Создать квадратную матрицу B=[a1 b1 c1; a2 b2 c2; a3 b3 c3]
размером 3х3. Вычислить определитель матрицы B.
1)по правилу Саррюса, обращаясь через индексы к элементам массива 2)разложить по первой строке, обращаясь через индексы к элементам массива
3)сделать проверку, обращаясь к стандартной функции det()
>> B=[a1 b1 c1; a2 b2 c2; a3 b3 c3]
B =
[ a1, b1, c1]
[ a2, b2, c2]
[ a3, b3, c3]
>> b=B(1,1)*B(2,2)*B(3,3)+B(2,1)*B(3,2)*B(1,3)+B(1,2)*B(2,3)*B(3,1)-B(3,1)*B(2,2)*B(1,3)...
-B(3,2)*B(2,3)*B(1,1)-B(2,1)*B(1,2)*B(3,3)
b =
a1*b2*c3+a2*b3*c1+b1*c2*a3-a3*b2*c1-b3*c2*a1-a2*b1*c3
>> S1=[B(2,2) B(3,2); B(2,3) B(3,3)]
S1 =
[ b2, b3]
[ c2, c3]
>> S2=[B(2,1) B(3,1); B(2,3) B(3,3)]
S2 =
[ a2, a3]
[ c2, c3]
>> S3=[B(2,1) B(2,2); B(3,1) B(3,2)]
S3 =
[ a2, b2]
[ a3, b3]
>> d=B(1,1)*det(S1)-B(1,2)*det(S2)+B(1,3)*det(S3
d =
a1*(b2*c3-b3*c2)-b1*(a2*c3-a3*c2)+c1*(a2*b3-a3*b2)
>> clear all
>> A=[1 2 3; 4 5 6; 7 8 1]
A =
1 2 3
4 5 6
7 8 1
>> S1=[A(2,2) A(3,2); A(2,3) A(3,3)]
S1 =
5 8
6 1
>> S2=[A(2,1) A(3,1); A(2,3) A(3,3)]
S2 =
4 7
6 1
>> S3=[A(2,1) A(2,2); A(3,1) A(3,2)]
S3 =
4 5
7 8
>> d=A(1,1)*det(S1)-A(1,2)*det(S2)+A(1,3)*det(S3)
d =
24
>> A=[3 4 -5; 8 7 -2; 2 -1 8]
A =
3 4 -5
8 7 -2
2 -1 8
>> S1=[A(2,2) A(3,2); A(2,3) A(3,3)]
S1 =
7 -1
-2 8
>> S2=[A(2,1) A(3,1); A(2,3) A(3,3)]
S2 =
8 2
-2 8
>> S3=[A(2,1) A(2,2); A(3,1) A(3,2)]
S3 =
8 7
2 -1
>> d=A(1,1)*det(S1)-A(1,2)*det(S2)+A(1,3)*det(S3)
d =
0
>> B=[1 2 3; 4 5 6; 7 8 1]
B =
1 2 3
4 5 6
7 8 1
>> S1=[B(2,2) B(3,2); B(2,3) B(3,3)]
S1 =
5 8
6 1
>> S2=[B(2,1) B(3,1); B(2,3) B(3,3)]
S2 =
4 7
6 1
>> S3=[B(2,1) B(2,2); B(3,1) B(3,2)]
S3 =
4 5
7 8
>> d=B(1,1)*det(S1)-B(1,2)*det(S2)+B(1,3)*det(S3)
d =
24
>> d=B(1,1)*B(2,2)*B(3,3)+B(2,1)*B(3,2)*B(1,3)+B(1,2)*B(2,3)*B(3,1)-B(3,1)*B(2,2)*B(1,3)...
-B(3,2)*B(2,3)*B(1,1)-B(2,1)*B(1,2)*B(3,3)
d =
24
>> B=[3 4 -5; 8 7 -2; 2 -1 8]
B =
3 4 -5
8 7 -2
2 -1 8
>> S1=[B(2,2) B(3,2); B(2,3) B(3,3)]
S1 =
7 -1
-2 8
>> S2=[B(2,1) B(3,1); B(2,3) B(3,3)]
S2 =
8 2
-2 8
>> S3=[B(2,1) B(2,2); B(3,1) B(3,2)]
S3 =
8 7
2 -1
>> d=B(1,1)*det(S1)-B(1,2)*det(S2)+B(1,3)*det(S3)
d =
0
>> d=B(1,1)*B(2,2)*B(3,3)+B(2,1)*B(3,2)*B(1,3)+B(1,2)*B(2,3)*B(3,1)-B(3,1)*B(2,2)*B(1,3)...
-B(3,2)*B(2,3)*B(1,1)-B(2,1)*B(1,2)*B(3,3)
d =
0
>> syms a b c x
>> B=[a+x x x; x b+x x; x x c+x]
B =
[ a+x, x, x]
[ x, b+x, x]
[ x, x, c+x]
>> S1=[B(2,2) B(3,2); B(2,3) B(3,3)]
S1 =
[ b+x, x]
[ x, c+x]
>> S2=[B(2,1) B(3,1); B(2,3) B(3,3)]
S2 =
[ x, x]
[ x, c+x]
>> S3=[B(2,1) B(2,2); B(3,1) B(3,2)]
S3 =
[ x, b+x]
[ x, x]
>> d=B(1,1)*det(S1)-B(1,2)*det(S2)+B(1,3)*det(S3)
d =
(a+x)*(b*c+b*x+x*c)-x^2*c-x^2*b
>> d=B(1,1)*B(2,2)*B(3,3)+B(2,1)*B(3,2)*B(1,3)+B(1,2)*B(2,3)*B(3,1)-B(3,1)*B(2,2)*B(1,3)...
-B(3,2)*B(2,3)*B(1,1)-B(2,1)*B(1,2)*B(3,3)
d =
(a+x)*(b+x)*(c+x)+2*x^3-(b+x)*x^2-x^2*(a+x)-x^2*(c+x)
>>