We can do row operations without changing det(A)'But let's say I have an arbitrary upper triangular matrix U U = [ a a a 0 b b 0 0 c ] And I perform the following row operations on U to bring it to U′ 1 a R 1 → R 1 1 b R 2 → R 2 1 c R 3 → R 3 Then U; is: U ′ = [ 1 1 1 0 1 1 0 0 1 ] But now det(U)=abc and det(U′)=1, thus det ( U ) ≠ det ( U ′ )All I've done is perform row operations on U to bring it to U′, but by performing those row operations, their determinants lose equality. How can that be possible?So how is this seeming contradiction is resolved. I'm assuming that I must have some misconception either on row operations or on determinants.Furthermore on a deeper level, what geometric interpretation/meaning does scaling the rows as I've done bringing U to U′, have on the determinant? Since the determinants of U and U′ are obviously no longer equal, geometrically what is this scaling doing to the determinant?

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We can do row operations without changing det(A)&#039;But let&#039;s say I have an arbitrary upper triangular matrix U  U  =      [                            a                          a                          a                                      0                          b                          b                                      0                          0                          c                      ]  And I perform the following row operations on U to bring it to U′      1    a        R    1    →      R    1        1    b        R    2    →      R    2        1    c        R    3    →      R    3  Then U; is:      U    ′    =      [                            1                          1                          1                                      0                          1                          1                                      0                          0                          1                      ]  But now det(U)=abc and det(U′)=1, thus  det  (  U  )  ≠  det  (      U    ′    )All I&#039;ve done is perform row operations on U to bring it to U′, but by performing those row operations, their determinants lose equality. How can that be possible?So how is this seeming contradiction is resolved. I&#039;m assuming that I must have some misconception either on row operations or on determinants.Furthermore on a deeper level, what geometric interpretation/meaning does scaling the rows as I&#039;ve done bringing U to U′, have on the determinant? Since the determinants of U and U′ are obviously no longer equal, geometrically what is this scaling doing to the determinant?

Eden Bradshaw · Accepted Answer

The determinant is an alternating multilinear function on the rows (or the columns, as you wish) of the matrix. What interests us here is the multilinearity.So here,   det  (  U  )  =  det  (      R    1    ,      R    2    ,      R    3    ). The matrix U′ obtained as you propose has determinant   det  (            R      1        a    ,            R      2        b    ,            R      3        c    ), ie, by multilinearity,                    det        (                  U          ′                )                            =                  1          a                det        (                  R          1                ,                              R            2                    b                ,                              R            3                    c                )                                          =                  1                      a            b                          det        (                  R          1                ,                  R          2                ,                              R            3                    c                )                                          =                  1                      a            b            c                          det        (                  R          1                ,                  R          2                ,                  R          3                )                                          =                  1                      a            b            c                          det        (        U        )            As @quid mentions in his comment, row operations are about adding a multiple of a row to another row. This operation does not affect the determinant because of its alternating and multilinear behaviour:  det  (      R    1    ,      R    2    +  a      R    1    ,      R    3    )  =  det  (      R    1    ,      R    2    ,      R    3    )  +  a  det  (      R    1    ,      R    1    ,      R    3    ) by linearity, and since it is alternating,   det  (      R    1    ,      R    1    ,      R    3    )  =  0 so   det  (      R    1    ,      R    2    +  a      R    1    ,      R    3    )  =  det  (      R    1    ,      R    2    ,      R    3    )Edit: Geometrically speaking, the determinant is the hypervolume of the       n          t      h      -dimensional parallelogram generated by the rows of your matrix, taken as vectors. So, if your matrix is not invertible, ie its rows are linearly dependent, then the rows form a parallelogram of volume 0, and so on.For instance, for a   2  ×  2 matrix, the determinant is the volume (=surface) of the parallelogram generated by your two rows. If these rows are linearly dependent, you can see that your parallelogram is flat, so has a volume (=surface) of 0.Dividing a row by   a means, for the determinant, dividing one of the lengths of your parallelogram by   a, hence dividing its volume by   a.

We can do row operations without changing det(A)' But let's say I have an arbitrary upper triangula

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