1.4.3 The Geometrical Meaning of the Determinant of Transformation, and the Multiplication Theorem: From the considerations of the last section, we can deduce the geometrical meaning of the determinant of an affine transformation and, at the same time, an algebraic theorem on the multiplication of determinants.
Consider a plane triangle with vertices (0, 0), (x1, y1), (x2, y2) with the area (cf. 1.2.1)
We shall investigate the relationship between A and the area A' of its image obtained by means of a primitive affine transformation
The vertices of the image triangle have the co-ordinates (0, 0), (ax1 + by1, y1), (ax2 + by2, y2), whence
However, this determinant can be transformed by the theorems of 1.3.1 in the following way:
i.e.,
If we had taken the primitive transformation
we should have found in the same way that
We thus see that a primitive affine transformation has the effect of multiplying the area of a triangle by a constant independent of the triangle.If the vertex of the triangle lies at the origin, the same fact applies, by virtue of the general formula for the area. Since the general affine transformation can be formed by combining primitive transformations, the statement remains true for any affine transformation. In the case of an affine transformation, the ratio of the area of an image triangle to the area of the original triangle is constant and independent of the choice of triangle, depending only on the coefficients of the transformation. In order to find this constant ratio, we consider, in particular, the triangle with vertices (0,0), (1,0) and (0,1), with area ½ . Since the image of this triangle, according to the transformation
has the vertices (0, 0), (a, c), (b, d), its area is
and we thus see that the constant ratio of area A'/A. for an affine transformation is the determinant of the transformation.
We can proceed in exactly the same way for transformations in space. If we consider the tetrahedron with the vertices (0,0,0), (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and subject it to the primitive transformation
the image tetrahedron has the vertices (0, 0, 0), (ax1 + by1 + cz1, y1, z1), (ax2 + by2 + cz2, y2, z2), (ax3 + by3 + cz3, y3 , z3), so that its volume is
Hence
where V is the volume of the original tetrahedron. For the volume of the image, given by the primitive transformation
we find in a similar manner that
and for the primitive transformation
that
Hence an arbitrary affine transformation has the effect of multiplying the volume of a tetrahedron by a constant. If no vertex of the tetrahedron coincides with the origin, this theorem follows from the general formula for the volume of a tetrahedron. In order to find this constant for the transformation
we consider the tetrahedron with the vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), the image of which has the vertices (0,0,0), (a,d,g), (b,e,h), (c,f,k). For the volumes V' and V of the image and the original we thus have
whence the determinant 
is the constant sought.
The sign of the determinant also has a geometrical meaning. In fact, from what we have seen in 2.3 regarding the connection between the sense of rotation and the volume of the tetrahedron or area of the triangle, it follows at once that a transformation with a positive determinant preserves the sense of rotation, while a transformation with a negative determinant reverses it.
We now consider the combination of two transformations
As we pass from x, y, z to x', y', z', the volume of a tetrahedron is multiplied by
as we pass from x', y', z to x", y", z", by
and by direct change from x, y, z to x", y", z", it is multiplied by
This observation yields the following relation, known as the theorem for the multiplication of determinants:
As before, we call the elements of the determinant on the
right hand side the products of the rows of
and the columns of
;
at the intersection of the i-th row and k-th
column of the product of the determinants, there stands the
expression formed from the i-th row of and
the k-th column of . Since rows and columns are
interchangeable, the product of the determinants can also be
obtained by combining columns and. rows, columns and columns, or
rows and. rows.
For two-rowed determinants, naturally, the corresponding theorem (combining rows and columns) holds:
1.Evaluate the determinants
2. Find the relation which must exist between a, b, c in order that the system of equations
may have a solution.
3*. (a) Prove the inequality
(b) When does the equality sign hold?
4. What conditions must be satisfied in order that the affine transformation
may leave the distance between any two points unchanged?
5. Prove that in an affine transformation the image of a quadratic
is again a quadratic.
6.* Prove that the affine transformation
leaves at least one direction unaltered.
7. Give the formulae for a rotation by the angle j about the axis x : y : z = 1 : 0 : -1 such that the rotation of the plane x == z is positive when looked at from the point (-1, 0, 1).
8. Prove that an affine transformation transforms the centre of mass of a system of particles into the centre of mass of the image particles.
9. If a1, ··· , g3 denote the quantities at the end of 1.2, defining a rotation of axes, then
