Mathematics with Maple

S.Duzhin

Solutions for Lesson 7

> with(linalg):

Exercise 1.

> A3:=matrix([[1,1/2],[-1/2,1]]);

> A4:=matrix([[1,0,0],[0,2,1],[0,0,3]]);

> A5:=matrix([[0,1,2],[1,2,3],[2,3,0],[3,0,1]]);

> multiply(A5,A4);

Exercise 2.

> S:=matrix([[0,-1],[1,0]]);

> T:=matrix([[1,1],[0,1]]);

> A1:=matrix([[1,3],[0,1]]);

> A2:=matrix([[1,-1],[0,1]]);

> TT:=multiply(T,T);

> TTT:=multiply(T,TT);

> evalm(TTT-A1);

> evalm(inverse(T)-A2);

> ST:=multiply(S,T);

> evalm(ST^6);

> inverseT:=multiply(evalm(ST^5),S);

> evalm(inverseT-A2);

Exercise 3.

> matrix([[a11,a12,a13,a14],[a21,a22,a23,a24],[a31,a32,a33,a34],[a41,a42,a43,a44]]);

> det(matrix([[a11,a12,a13,a14],[a21,a22,a23,a24],[a31,a32,a33,a34],[a41,a42,a43,a44]]));

Exercise 4.

> M:=matrix([[9, 7, -1, -5], [-22, -16, 2, 10], [18, 16, 0, -10], [-19, -13, 1, 7]]);

> p:=charpoly(M,x);

> factor(p);

> subs(x=M,p);

> evalm(%);

> q:=minpoly(M,x);

> factor(q);

> evalm(subs(x=M,q));

Exercise 5.

> matrix([[9, 7, -1, -5], [-22, -16, 2, 10], [18, 16, 0, -10], [-19, -13, 1, 7]]);

> eigenvals(M);

> eigenvects(M);

> v1:=vector([0,1,-1/2,3/2]);

> v2:=vector([1, 0, 1, 2]);

> v3:=vector([1,-2,-7, 0]);

> v4:=vector([0, 0,-5, 1]);

> evalm(multiply(M,v1)+scalarmul(v1,2));

> evalm(multiply(M,v2)+scalarmul(v2,2));

> evalm(multiply(M,v3)+scalarmul(v3,-2));

> evalm(multiply(M,v4)+scalarmul(v4,-2));

Exercise 6.

> M:=matrix([[1/2,1/2,1/2,1/2],[1/2,1/2,-1/2,-1/2],[1/2,-1/2,1/2,-1/2],[1/2,-1/2,-1/2,1/2]]);

> multiply(M,transpose(M));

Exercise 7.

> A:=matrix([[sqrt(3)/2,1/2],[-1/2,sqrt(3)/2]]);

> evalm(A^12);

Exercise 8.

> V3 := vandermonde([x1,x2,x3]);

> V4 := vandermonde([x1,x2,x3,x4]);

> factor(det(V3));

> factor(det(V4));