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Problem 2

Problem 2a

Definition of the arrays

This command creates a 3x3 matrix full of zeros. This is equivalent to the command DIMENSION in fortran, for

example.

delta = MatrixForm@Array@ - &, 83, 3

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epsilon@@1, 2, 3DD = 1;epsilon@@3, 1, 2DD = 1;epsilon@@2, 3, 1DD = 1;epsilon@@1, 3, 2DD = - 1;epsilon@@2, 1, 3DD = - 1;epsilon@@3, 2, 1DD = - 1;

MatrixForm@epsilonD

0

0

0

0

0

1

0

-1

0

0

0

-1

0

0

0

1

0

0

0

1

0

-1

0

0

0

0

0

The following do loops will do the requested sum and put the result in the variable b

b = 0.;

Do@Do@

Do@b = b + epsilon@@i, j, kDD * epsilon@@k, j, iDD,8k, 3

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MatrixForm@epsilonD

0

0

0

0

0

1

0

-1

0

0

0

-1

0

0

0

1

0

0

0

1

0

-1

0

0

0

0

0

The generic vector a will be defined directly in the next step

a = 8a1, a2, a3

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MatrixForm@epsilonD

0

0

0

0

0

1

0

-1

0

0

0

-1

0

0

0

1

0

0

0

1

0

-1

0

0

0

0

0

This command creates a 3x3 matrix full of zeros

delta = MatrixForm@Table@0, 8i, 3

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epsilon = MatrixForm@Table@0, 8i, 3

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e = 0;

Do@Do@

Do@Do@

e = e + epsilon@@i, k, sDD * epsilon@@m, k, sDD * delta@@i, mDD,8s, 3

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Tr@SmD

300.

- 555.5550903965119` + - 19.67149390163511` + 875.2265842981469`

300.

The second invariant is the sum of the principal minors, and the equality proposed in the problem statement also holds.The code shown below will calculate the sum of all subdeterminants obtained by removing the row and column of the

each of the diagonal components of the matrix

Det@Table@Sm@@i, jDD, 8i, 2, 3

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