Matrices and determinants appear in two other important contexts; one is in solving simultaneous linear equations in several variables. The other is in representing linear transformations of vectors. The first of these is discussed in detail in Chapter 32.
In the latter context a matrix represents the transformation that takes the column basis vectors into the vectors that are the corresponding columns of the matrix.
Sums of original basis vectors are transformed into the same sums of the corresponding columns. This fact defines the transformation on all vectors.
When the determinant of a matrix is zero, the volume of the region with sides given by its columns or rows is zero, which means the matrix considered as a transformation takes the basis vectors into vectors that are linearly dependent and define 0 volume.
This happens, the determinant is zero, when the columns (and rows) of the matrix are linearly dependent.
4.2 What is the matrix of the transformation which takes a unit vector in the direction of the x axis into one in the direction of the y axis, and similarly one along the y axis into one along the z axis, and one along the z axis into one along the x axis?
4.3 What matrix describes the transformation which doubles the component of vectors in the x direction, halves components in the y direction, and leaves components in the z direction alone.
4.4 What matrix describes the transformation in three dimensions which projects vectors into the (x, y) plane? onto the x axis? onto the diagonal in the (x, y) plane?