For each of the matrices in Exercise 1, use the XDX^{-1} factorization to compute A^6. Reference: Exercise 1: In each of the following, factor the matrix A into a product XDX^{-1}, where D is diagonal: (a) A = egin{pmatrix} 0 & 1 \ 1 & 0 end{pmatrix} (b) A = egin{pmatrix} 5 & 6 \ -2 & -2 end{pmatrix} (c) A = egin{pmatrix} 2 & -8 \ 1 & -4 end{pmatrix} (d) A = egin{pmatrix} 2 & 2 & 1 \ 0 & 1 & 2 \ 0 & 0 & -1 end{pmatrix} (e) A = egin{pmatrix} 1 & 0 & 0 \ -2 & 1 & 3 \ 1 & 1 & -1 end{pmatrix} (f) A = egin{pmatrix} 1 & 2 & -1 \ 2 & 4 & -2 \ 3 & 6 & -3 end{pmatrix}
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For matrix A = (9 0, -8 2), the characteristic equation is: det(A - λI) = det((9-λ) 0, -8 (2-λ)) = (9-λ)(2-λ) = 0 Solving for λ, we get λ = 9, 2 ** Show more…
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