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https://texercises.com/exercise/trace-and-characteristic-polynomials/
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Exercise:
Show that c_n--^n- textTrA c_textdetA In other words P_Ax-^nx^n+-^n-textTrA x^n-+c_n-x^n-+...+c_x+textdetA

Solution:
Proof. P_Axtextdet pmatrix A_-x & A_ & hdots & A_n A_ & A_-x & hdots & A_n vdots & vdots & vdots & vdots A_n & A_n & hdots & A_nn-x pmatrix A_-x A_-x ... A_nn-x+_sigmain S_n sigma neq idtextsgnsigmaB_sigma_ ... B_nsigma_n *. If sigmanneq id then exists at least two indices ineq j s.t. ineq sigma i and jneq sigma j Longrightarrow the product * contains at least two factors B_i sigma i and B_j sigma j that are not from the diagonal of B and so do not contain x Longrightarrow textdet*leq n- Longrightarrow P_AxA_-x ... A_nn-x+texta polynom of degleg n- -^nx^n+A_-x^n-+...+A_nn-x^n-+texta polynom of degleg n- -^nx^n+-^n- textTrAx^n-+...+c_x+c_. Finally c_P_AtextdetA- ItextdetA.
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Exercise:
Show that c_n--^n- textTrA c_textdetA In other words P_Ax-^nx^n+-^n-textTrA x^n-+c_n-x^n-+...+c_x+textdetA

Solution:
Proof. P_Axtextdet pmatrix A_-x & A_ & hdots & A_n A_ & A_-x & hdots & A_n vdots & vdots & vdots & vdots A_n & A_n & hdots & A_nn-x pmatrix A_-x A_-x ... A_nn-x+_sigmain S_n sigma neq idtextsgnsigmaB_sigma_ ... B_nsigma_n *. If sigmanneq id then exists at least two indices ineq j s.t. ineq sigma i and jneq sigma j Longrightarrow the product * contains at least two factors B_i sigma i and B_j sigma j that are not from the diagonal of B and so do not contain x Longrightarrow textdet*leq n- Longrightarrow P_AxA_-x ... A_nn-x+texta polynom of degleg n- -^nx^n+A_-x^n-+...+A_nn-x^n-+texta polynom of degleg n- -^nx^n+-^n- textTrAx^n-+...+c_x+c_. Finally c_P_AtextdetA- ItextdetA.
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characteristic polynomial, eth, fs23, lineare algebra, proof
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(3, default)
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ENG (English)
Type
Proof
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