I'm curious about is there any geometric property relative to negative value for determinant of matr

Brunton39

Brunton39

Answered question

2022-06-23

I'm curious about is there any geometric property relative to negative value for determinant of matrix.
det ( A ) < 0
I knew about some of determinant of matrix properties as following, but it seems to me that it is nothing relative to negative value of determinant of matrix
det ( A B ) = det ( A ) d e t ( B ) ( M u l t i p l i c a t i v e )
det ( A ) = 0 A is singular
M 2 , 2 = [ a b c d ]
| det ( M 2 × 2 ) | = | a d b d | = volumn of parallelogram
| det ( M n × n ) | = j = 1 n a i , j ( 1 ) i + j det ( M i , j ) expansion of determinant alone the  i t h  row

Answer & Explanation

aletantas1x

aletantas1x

Beginner2022-06-24Added 22 answers

The geometric property of ± 1 signed determinant matrices is the orientation of the vectors of the matrix in its given space(for e.g R n ) with respect to a fixed orientation of the basis vectors. If the vectors of your matrix A has the same orientation as the basis then s i g n ( d e t ( A ) ) = + 1 otherwise its 1.

For an example, take the matrix A = ( e 2 , e 1 , e 3 ) = ( ( 0 , 1 , 0 ) , ( 1 , 0 , 0 ) , ( 0 , 0 , 1 ) ) in R 3 . Then think of the orientation of the standard basis axes, ( e 1 , e 2 , e 3 ), in R 3 to be c l o c k w i s e such that if you drew an equilateral triangle in the R 3 plane intersecting ( e 1 , e 2 , e 3 ), then to get from e 1 to e 2 to e 3 you would have to travel around the triangle in a clockwise rotation. If you perform the same triangle exercise with the vector ( e 2 , e 1 , e 3 ) you would see that you move c o u n t e r c l o c k w i s e around the triangle. Thus, the matrix A = ( e 2 , e 1 , e 3 ) has the opposite orientation as the standard basis and s i g n ( d e t ( A ) ) = 1.

This also explains why when you swap two columns/rows of a matrix the sign of the determinant changes with each swap. Unfortunately though, this geometric idea is a bit more difficult to understand in higher dimensions.

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