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Hello,

I need some help in the fist parts of two lineal algebra problems, specially with algebraic manipulation. I guess that if I rewrite the determinant nicely some terms get canceled and I can write the inverse nicely, but don't know how to do it...

Problem 1:

Problem 2:

3. The attempt at a solution

Problem 1:

(1) [tex] Det(A) = a(a^2-b^2) -b(ba-b^2) + b(b^2-ab) = b(b-a)(2b-\frac{a^2}{b})[/tex]

(2) [tex] A^-1 = \frac {adj(A)}{Det(A)} [/tex]

[tex]Adj(A) = \left[ \begin{matrix} a^2-b^2 & b(b-a) & b(b-a) \\ b(b-a) & a^2-b^2 & b(b-a) \\ b(b-a) & b(b-a) & a^2-b^2 \end{matrix} \right][/tex]

[tex] A^-1 = ? [/tex]

(I can write the terms outside the diagonal nicely because some parts get cancelled, but not the diagonal itself...)

Problem 2:

(1)

Sum of eigenvalues:

Trace(A) = a + b + c

Product of eigenvalues:

[tex] Det(A) = a(bc -b^2) -a(ac-ab) + 0 = abc - ab^2 - a^2c + a^2b [/tex]

(2) [tex] A^-1 = \frac {adj(A)}{Det(A)} [/tex]

[tex]Adj(A) = \left[ \begin{matrix} b(c-b) & a(b-c) & 0 \\ a(b-c) & a(c-a) & a(a-b) \\ 0 & a(a-b) & a(b-a) \end{matrix} \right][/tex]

[tex] A^-1 = ? [/tex]

Thanks in advance...

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# Homework Help: Lineal Algebra: Inverse Matrix of Symmetric Matrix

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