Let A = (aij) ∈ M atK (n). A Tracef Definition is Tr(A) = a11 + a22 + · · · + ann (cross number of ads). a) Prove that PA(λ) = det(A − λI) = (−1) n (λ) n- Tr(A) n − 1 + · · · + (- 1) n det A). b) Prove det (P −1AP - λI) = PA(λ), for every P match. c) Prove Tr(A) = Tr(P −1AP), with every inverse matrix P. d) Suppose 1, · · ·, n K, are eigenvalues ​​of A. Calculate Tr(A) and det A through that private value

Consider the following geometry problems in 3-space enter t or f depending on whether the statement is true or false. (you must enter t or f -- true and false will not work.) f equation editorequation editor 1. a plane and a line either intersect or are parallel t equation editorequation editor 2. two planes parallel to a third plane are parallel f equation editorequation editor 3. two lines either intersect or are parallel f equation editorequation editor 4. two planes orthogonal to a third plane are parallel f equation editorequation editor 5. two planes orthogonal to a line are parallel f equation editorequation editor 6. two planes either intersect or are parallel f equation editorequation editor 7. two lines orthogonal to a third line are parallel t equation editorequation editor 8. two lines parallel to a third line are parallel t equation editorequation editor 9. two planes parallel to a line are parallel f equation editorequation editor 10. two lines parallel to a plane are parallel f equation editorequation editor 11. two lines orthogonal to a plane are parallel