- #1
degs2k4
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Homework Statement
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:
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...