Exercise
https://texercises.com/exercise/determinant-multiplication/
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.

Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
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.
Report An Error
You are on texercises.com.
reCaptcha will only work on our main-domain \(\TeX\)ercises.com!
Meta Information
\(\LaTeX\)-Code
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.
Contained in these collections

Similar exercises (48)
Title Creator Matched on
Determinants and transposition rk tags
Determinants of block matrices rk tags
Determinants and row operations rk tags
Determinant function rk tags
Determinants and cofactors rk tags
more (43 more)
Attributes & Decorations
Tags
determinant, eth, fs23, lineare algebra, proof
Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
Decoration
Content image