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I need help on these questions for an assignment. I've been working on them for a couple of days and not getting anywhere. Any help would be appreciated...

1) A certain 4X4 real matrix is known to have these properties:

1. Two fo the eigenvalues of A are 3 and 2

2. the number 3 is an eigenvalue of the matrix A + 2I

3. det A =12

(i) what are the other two eigenvalues of the A?

(ii) what is the characteristic polynomial of A? of A'

(iii)what is the characteristic polynomial of A^-1

i guess (ii) and (iii) are easy once you get (i)

2) Let T(n) denote an nXn matrix such that for each tij,

tij= a if i=j

tij=b if i not equals j

(so basically a matrix with a's in the diagnol and b's as all the other elements

eg 3X3 a b b

b a b

b b a

(i) verify that T(n) = (a-b)I + bE where E is an nXn matirx of all 1's

(ii)find the eigenvalues of T

(iii) show that det T(n) = (a-b)^n-1 * (a + (n-1)b)

Thanks :)

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# 4X4 real matrix eigenvalues

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