Matrices
- Let
- Find
- Find
- Find
times its
inverse - Find
- Find
the transpose of
- Find
the transpose of
- Find
- Find
- Find
- Find
- Evaluate
the determinant of the transpose of
- Multiply
the determinant of the inverse of times
- Find
all eigenvalues and corresponding eigenvectors of the given matrices.
Check by substituting each eigenvalue and the corresponding eigenvector in
the matrix equation
Solutions
The following is the MAPLE sheet to solve these two examples.
> with(linalg);
Warning, the protected names norm and trace have been redefined and unprotected
> A:=matrix(3,3,[2,-3,2,0,-1,7,6,-1,1]);
![A := matrix([[2, -3, 2], [0, -1, 7], [6, -1, 1]])](file:///E:\..\..\matrices11.gif)
> B:=matrix(3,3,[-1,1,5,2,-3,1,9,0,-2]);
![B := matrix([[-1, 1, 5], [2, -3, 1], [9, 0, -2]])](file:///E:\..\..\matrices12.gif)
> C:=matrix(3,1,[5,1,3]);
![C := matrix([[5], [1], [3]])](file:///E:\..\..\matrices13.gif)
> X:=matrix(3,1,[x^2,ln(x),(1-x)]);
![X := matrix([[x^2], [ln(x)], [1-x]])](file:///E:\..\..\matrices14.gif)
> evalm(2*B);
![matrix([[-2, 2, 10], [4, -6, 2], [18, 0, -4]])](file:///E:\..\..\matrices15.gif)
> #b]
> V:=multiply(A,B);
![V := matrix([[10, 11, 3], [61, 3, -15], [1, 9, 27]]...](file:///E:\..\..\matrices16.gif)
> #c]
> multiply(A,inverse(A));
![matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])](file:///E:\..\..\matrices17.gif)
> #d]
> inverse(V);
![matrix([[-18/1207, 45/2414, 29/2414], [277/2414, -8...](file:///E:\..\..\matrices18.gif)
> transpose(B);
![matrix([[-1, 2, 9], [1, -3, 0], [5, 1, -2]])](file:///E:\..\..\matrices19.gif)
> transpose(multiply(B,C));
![matrix([[11, 10, 39]])](file:///E:\..\..\matrices20.gif){kind=small}
> #g]
> evalm(-10*V);
![matrix([[-100, -110, -30], [-610, -30, 150], [-10, ...](file:///E:\..\..\matrices21.gif)
> #h]
> evalm(A^3);
![matrix([[-58, -28, -18], [84, -120, 42], [72, -42, ...](file:///E:\..\..\matrices22.gif)
> #j]
> matadd(multiply(A,C),X);
![matrix([[13+x^2], [20+ln(x)], [33-x]])](file:///E:\..\..\matrices23.gif)
> #i]
> evalm(-10*multiply(A,X));
![matrix([[-20*x^2+30*ln(x)-20+20*x], [-70+10*ln(x)+7...](file:///E:\..\..\matrices24.gif)
> #k]
> det(transpose(B));
> #l]
> evalm(det(inverse(evalm(A^3)))*B);
![matrix([[1/1061208, -1/1061208, -5/1061208], [-1/53...](file:///E:\..\..\matrices26.gif)
>
> #EXample 2
> A:=matrix(2,2,[6,-5,-2,9]);
![A := matrix([[6, -5], [-2, 9]])](file:///E:\..\..\matrices27.gif)
> eigenvects(A);
![[11, 1, {vector([-1, 1])}], [4, 1, {vector([5/2, 1]...](file:///E:\..\..\matrices28.gif){kind=small}
> v1:=matrix(2,1,[1,-1]);
![v1 := matrix([[1], [-1]])](file:///E:\..\..\matrices29.gif)
> v2:=matrix(2,1,[5/2,1]);
![v2 := matrix([[5/2], [1]])](file:///E:\..\..\matrices30.gif)
> evalm(A&*v1=11*v1);
![matrix([[11], [-11]]) = matrix([[11], [-11]])](file:///E:\..\..\matrices31.gif)
> evalm(A&*v2=4*v2);
![matrix([[10], [4]]) = matrix([[10], [4]])](file:///E:\..\..\matrices32.gif)
> #b]
> A:=matrix(3,3,[-4,0,6,2,4,-4,2,0,7]);
![A := matrix([[-4, 0, 6], [2, 4, -4], [2, 0, 7]])](file:///E:\..\..\matrices33.gif)
> eigenvects(A);
![[-5, 1, {vector([-6, 16/9, 1])}], [4, 1, {vector([0...](file:///E:\..\..\matrices34.gif){kind=small}
> v1:=matrix(3,1,[-6,16/9,1]):v2:=matrix(3,1,[0,1,0]):v3:=matrix(3,1,[1,-3/2,2]):
> evalm(A&*v1=-5*v1);
![matrix([[30], [-80/9], [-5]]) = matrix([[30], [-80/...](file:///E:\..\..\matrices35.gif)
> evalm(A&*v2=4*v2);
![matrix([[0], [4], [0]]) = matrix([[0], [4], [0]])](file:///E:\..\..\matrices36.gif)
> evalm(A&*v3=8*v3);
![matrix([[8], [-12], [16]]) = matrix([[8], [-12], [1...](file:///E:\..\..\matrices37.gif)
> #c]
> A:=matrix(3,3,[2,-3,1,2,-3,0,6,-45,12]);
![A := matrix([[2, -3, 1], [2, -3, 0], [6, -45, 12]])...](file:///E:\..\..\matrices38.gif)
> charpoly(A,lambda);
> eigenvects(A);
![[-3, 1, {vector([0, 1, 3])}], [2, 1, {vector([5/2, ...](file:///E:\..\..\matrices40.gif){kind=small}
Ø # the rest of the example is left as an exercise.
Ø