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Exercise:
forall AB in M_ntimes nK we have textdetA BtextdetA textdetB.

Solution:
Proof. Fix a matrix Bin M_ntimes nK and consider a function D:M_ntimes nKrightarrow K defined by DA:textdetA B. bf claim: D is n-linear and alternating. Proof of claim. Write A pmatrix hdots & alpha_ & hdots hdots & vdots & hdots hdots & alpha_n & hdots pmatrix. If we view alpha_i as a times n matrix then alpha_i B is also times n and in fact A B pmatrix hdots & alpha_ B & hdots hdots & vdots & hdots hdots & alpha_i B & hdots hdots & vdots & hdots hdots & alpha_n B & hdots pmatrix. So DAtextdetalpha_ B...alpha_n B. Clearly forall two row vectors alpha_i alpha_i': alpha_i+alpha_i' B alpha_i B+alpha_i' B and forall cin K: calpha_i B c alpha_i B. Since det is n-linear it follows that D is also n-linear. If alpha_ialpha_j Longrightarrow alpha_i B alpha_j B Longrightarrow DAtextdetalpha_ B...alpha_n B This proves the claim. We continue with the proof of the theorem. By a previous theorem DAtextdetA DI. But DItextdetI B textdetB.
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Exercise:
forall AB in M_ntimes nK we have textdetA BtextdetA textdetB.

Solution:
Proof. Fix a matrix Bin M_ntimes nK and consider a function D:M_ntimes nKrightarrow K defined by DA:textdetA B. bf claim: D is n-linear and alternating. Proof of claim. Write A pmatrix hdots & alpha_ & hdots hdots & vdots & hdots hdots & alpha_n & hdots pmatrix. If we view alpha_i as a times n matrix then alpha_i B is also times n and in fact A B pmatrix hdots & alpha_ B & hdots hdots & vdots & hdots hdots & alpha_i B & hdots hdots & vdots & hdots hdots & alpha_n B & hdots pmatrix. So DAtextdetalpha_ B...alpha_n B. Clearly forall two row vectors alpha_i alpha_i': alpha_i+alpha_i' B alpha_i B+alpha_i' B and forall cin K: calpha_i B c alpha_i B. Since det is n-linear it follows that D is also n-linear. If alpha_ialpha_j Longrightarrow alpha_i B alpha_j B Longrightarrow DAtextdetalpha_ B...alpha_n B This proves the claim. We continue with the proof of the theorem. By a previous theorem DAtextdetA DI. But DItextdetI B textdetB.
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determinant, eth, fs23, lineare algebra, proof
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(3, default)
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ENG (English)
Type
Proof
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