How can a determinant of a matrix become an area?

[RyozKidz]

can anyone explain or prove this??

Ax={(x^TA^T)^T}

how can a determinant of a matrix become an area??

example: 2 X 2 matrix

the determinant of this matrix is ad-bc !
but i search on wikipedia it wrote like this :The assumption here is that a linear transformation is applied to row vectors as the vector-matrix product xTAT, where x is a column vector. The parallelogram in the figure is obtained by multiplying matrix A (which stores the co-ordinates of our parallelogram) with each of the row vectors [0,0] [1,0] [1,1] [0,1]in turn. These row vectors define the vertices of the unit square.

 

[maze]

The action of a matrix on vectors in the plane is to map squares to parallelograms. The parallelograms may be skewed, rotated, scaled, flipped, etc by the matrix. The determinant is the ratio of the new parallelogram area to the original square area. If the square gets flipped or turned "inside out", then it is as if the parallelogram has "negative" area, so the determinant is given a minus sign.

In 3 dimension, a matrix maps cubes to parallelipipeds, and the determinant is the ratio of the volumes.

You can actually define determinants in this way, think about how swapping and scaling rows affects volumes, and then derrive the determinant formula via row operations. However most courses go the other way around and start off with the formula, then show it is the volume factor.