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Matrices, Eigenvectors, Eigenvalues and Inverses

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Determining the equation for a matrix and confirming the inverse, eigenvalues
I have think the answer to question (a) is theta^2 [(1-p)I + pJ]. But when I am multiplying the sigma and the sigma inverse I still have "stuff" at the tail end of the identity......being added. I know I should be left with the identity if the two pieces are the inverses to each other.

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Solution Summary

Matrices, Eigenvectors, Eigenvalues and Inverses are investigated.

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Please see the attached file.

The attached document gives an explanation of the inverse verification, and demonstrates the eigenvalues and eigenvectors for sigma.

Part (a)

To show that is the inverse, we will verify that:

We can consider first the diagonal entries and off-diagonal entries separately. If we can prove that:

(1) all diagonal entries in the resulting matrix are 1.
(2) all off-diagonal entries in the resulting matrix are 0.

We will have shown that .

(1) Diagonal entries in the above matrix have the form:

This verifies that all the diagonal entries are indeed 1.

(2) The ...

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