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https://texercises.com/exercise/determinants-and-row-operations/
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Exercise:
abcliste abc If B is obtained from A by adding a multiple of one row of A to another row or by doing the analogous operation on columns then textdetBtextdetA. R_i+c R_jlongrightarrow R_i ineq j cin K then textdetBtextdetA. abc If R_i longleftrightarrow R_j ineq j then textdetB-textdetA abc If c R_ilongrightarrow R_i then textdetBctextdetA. abcliste

Solution:
Proof. b follows from the fact that det is alternating. c follows from n-linearity of det. a Write A pmatrix hdots & alpha_ & hdots hdots & vdots & hdots hdots & alpha_n & hdots pmatrix. Ase j i. textdetBtextdetalpha_...alpha_i+c alpha_j...alpha_j...alpha_j...alpha_n textdetalpha_...alpha_n+c textdetalpha_...alpha_j...alpha_j...alpha_j...alpha_n textdetA+ If j i the proof is similar.
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Exercise:
abcliste abc If B is obtained from A by adding a multiple of one row of A to another row or by doing the analogous operation on columns then textdetBtextdetA. R_i+c R_jlongrightarrow R_i ineq j cin K then textdetBtextdetA. abc If R_i longleftrightarrow R_j ineq j then textdetB-textdetA abc If c R_ilongrightarrow R_i then textdetBctextdetA. abcliste

Solution:
Proof. b follows from the fact that det is alternating. c follows from n-linearity of det. a Write A pmatrix hdots & alpha_ & hdots hdots & vdots & hdots hdots & alpha_n & hdots pmatrix. Ase j i. textdetBtextdetalpha_...alpha_i+c alpha_j...alpha_j...alpha_j...alpha_n textdetalpha_...alpha_n+c textdetalpha_...alpha_j...alpha_j...alpha_j...alpha_n textdetA+ If j i the proof is similar.
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Tags
determinant, eth, fs23, lineare algebra, proof
Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
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