Question

What is the physical meaning of determinant of a matrix?

Answer 1

Each  n×nn×n  matrix describes a linear transformation  T:Rn→Rn.T:Rn→Rn.  If you take a figure S⊆Rn,S⊆Rn,  then that transformation maps it to its image, another figure T(S)⊆Rn.T(S)⊆Rn.   

In the case when the transformation preserves orientation, the n-dimensional content (length when n = 1, area when n = 2, volume when n = 3, etc.) will scale by a factor of the determinant of the matrix; the content of T(S)T(S) will be the determinant times the content of S.S.  For an orientation reversing transformation the factor is the negation of the determinant.

Consider, for example, the 2×22×2 matrix

                                            [03/2−2/30][0−2/33/20]

That matrix describes a linear transformation which is the composition of stretching vertically by a factor of 3/2, squeezing horizontally by a factor of 2/3, then rotating counterclockwise by 90°.  Its determinant is 1.  If you start with an initial figure, it will be stretched, squeezed, and rotated, but since the determinant of the matrix is 1, the resulting figure will have the same area as the original one.

Answer 2

Answer:

In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. ... Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix.