Matrices

 

  1. Let

    1. Find
    2. Find
    3. Find times its inverse
    4. Find
    5. Find the transpose of
    6. Find the transpose of
    7. Find
    8. Find
    9. Find
    10. Find
    11. Evaluate the determinant of the transpose of
    12. Multiply the determinant of the inverse of times
  1. 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.

 

> #Example 1:#a]

> with(linalg);

Warning, the protected names norm and trace have been redefined and unprotected

 

[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...
[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp,...

> A:=matrix(3,3,[2,-3,2,0,-1,7,6,-1,1]);

A := matrix([[2, -3, 2], [0, -1, 7], [6, -1, 1]])

> B:=matrix(3,3,[-1,1,5,2,-3,1,9,0,-2]);

B := matrix([[-1, 1, 5], [2, -3, 1], [9, 0, -2]])

> C:=matrix(3,1,[5,1,3]);

C := matrix([[5], [1], [3]])

> X:=matrix(3,1,[x^2,ln(x),(1-x)]);

X := matrix([[x^2], [ln(x)], [1-x]])

> evalm(2*B);

matrix([[-2, 2, 10], [4, -6, 2], [18, 0, -4]])

> #b]

> V:=multiply(A,B);

V := matrix([[10, 11, 3], [61, 3, -15], [1, 9, 27]]...

> #c]

> multiply(A,inverse(A));

matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])

> #d]

> inverse(V);

matrix([[-18/1207, 45/2414, 29/2414], [277/2414, -8...

> transpose(B);

matrix([[-1, 2, 9], [1, -3, 0], [5, 1, -2]])

> transpose(multiply(B,C));

matrix([[11, 10, 39]])

> #g]

> evalm(-10*V);

matrix([[-100, -110, -30], [-610, -30, 150], [-10, ...

> #h]

> evalm(A^3);

matrix([[-58, -28, -18], [84, -120, 42], [72, -42, ...

> #j]

> matadd(multiply(A,C),X);

matrix([[13+x^2], [20+ln(x)], [33-x]])

> #i]

> evalm(-10*multiply(A,X));

matrix([[-20*x^2+30*ln(x)-20+20*x], [-70+10*ln(x)+7...

> #k]

> det(transpose(B));

142

> #l]

> evalm(det(inverse(evalm(A^3)))*B);

matrix([[1/1061208, -1/1061208, -5/1061208], [-1/53...

>

> #EXample 2

> A:=matrix(2,2,[6,-5,-2,9]);

A := matrix([[6, -5], [-2, 9]])

> eigenvects(A);

[11, 1, {vector([-1, 1])}], [4, 1, {vector([5/2, 1]...

> v1:=matrix(2,1,[1,-1]);

v1 := matrix([[1], [-1]])

> v2:=matrix(2,1,[5/2,1]);

v2 := matrix([[5/2], [1]])

> evalm(A&*v1=11*v1);

matrix([[11], [-11]]) = matrix([[11], [-11]])

> evalm(A&*v2=4*v2);

matrix([[10], [4]]) = matrix([[10], [4]])

> #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]])

> eigenvects(A);

[-5, 1, {vector([-6, 16/9, 1])}], [4, 1, {vector([0...

> 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/...

> evalm(A&*v2=4*v2);

matrix([[0], [4], [0]]) = matrix([[0], [4], [0]])

> evalm(A&*v3=8*v3);

matrix([[8], [-12], [16]]) = matrix([[8], [-12], [1...

> #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]])...

> charpoly(A,lambda);

(lambda-2)*(lambda+3)*(lambda-12)

> eigenvects(A);

[-3, 1, {vector([0, 1, 3])}], [2, 1, {vector([5/2, ...

       # the rest of the example is left as an exercise.

        

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