Determinant of transpose | Matrix transformations | Linear Algebra | Khan Academy

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Proof by induction that transposing a matrix does not change its determinant
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[@ericgc01]

This proof was clearer than the one in my textbook. Thank you.

[@davidmurphy563]

I wish we had a graphical representation of the det, at least of identity in R2 and R3. It just seems self evident that if you flip the basis then the area/volume swept will remain unchanged. I don't know why we need a proof for something like that.

[@rohitg1529]

While this gives a good idea for why the determinant is invariant under transposition, the proof itself is circular. We use this result in deriving the formula for the determinant of a matrix from its sub-matrices or cofactors. So the formula cannot be used in proving this result.

[@megamewtwoy9316]

plz bro tell how to prove it i want a strong proof of this property

[@smithdb08]

Thank you so much....now I have a better understanding of this particular proof in addition to a better understanding of inductive proofs!

[@alkalait]

i think it's worth doing a video dedicated to proof by induction with more examples

[@iveshoot]

If you assume the Laplace expansion works on both row and column no induction is needed

[@TehRenting]

so much more clear than in my courses serliously thanks ALOT!!!

[@avichalpratap3425]

This is very helpful video for 11th and 12th class students.And i give special thanks to Sal sir for making great video.

[@joash480]

holy hell I don't get any of it :|

[@EclecticSceptic]

The proof doesn't even need to be this complicated. If you look at the definition of the determinant it is very simple. When you see the bi-vector, tri-vector, etc. definition, you see that the determinant can be taken by expanding out the top row, or expanding the 1st column, or any other column or row. It is a symmetric and wonderful expression! It also explains why swapping rows makes the determinant negative. /watch?v=6XghF70fqkY&list=PL01A21B9E302D50C1&index=4&feature=plpp_video

[@paradoxwastaken]

The link doesn’t work lol

[@roveenkumar8612]

@@paradoxwastaken 9 years ago

[@paradoxwastaken]

@@roveenkumar8612 and I wrote “the link doesn’t work lol” 2 months ago, I didn’t even read how long ago the original comment was made

[@mrsandman4915]

Thank you!

[@MrUltrablixten]

Would you not first have to prove that the determinant is linear and alternating with respect to its rows before doing the expansions which the induction proof relies on?

[@shineyoung1746]

国内讲行列式等于其转置的内容在计算行列式内容的前面，所以证明过程相当难，特别考验抽象思维能力。