
The first property, which we deduce from the definition of determinant and what we already know about areas and volumes, is the value of the determinant of an array with all its nonzero entries on the main diagonal. Such an array describes a figure which is a rectangle or rectangular parallelepiped, with sides that are parallel to the x and y and z and whatever axes. We already know that the magnitude of this determinant must be the product of its diagonal entries. The sign we define to be that of this product.
Thus the determinants of the three arrays above are 5, 1 and 2, respectively.
This is wonderful?
No, not yet. This is a definition of the sign of a determinant. It depends on the order in which you choose to list the sides of the figure.
We really are interested in the area of parallelograms that are tilted, so that sides are not perpendicular to one another, or that are rotated, so that the sides are not parallel to axes.
And here is the wonderful fact: If you fix the base of a parallelogram, (one side of it,) then its area only depends on the height of the parallelogram from that base. It does not matter how much the second side tilts in the direction of the base.
A similar property holds in any dimension, and tells us: we can add any multiple of one row of the array to any other row, without changing its determinant.
Another wonderful fact that follows from the first two is: the determinant is linear in any of the rows (or columns) of its array. This means that if you multiply some row by 7 the value of the determinant goes up by a factor of 7. It also means that if you take two arrays that differ only in some one row, like the following two, which differ only in their first rows:
then the determinant of the array gotten by summing the rows that differ and keeping the others the same, (getting here 3 4 for the first row and 5 7 for the second) is the sum of the determinants of the two arrays you started with.
Exercise 17.5 Show, by adding rows to one another appropriately, that interchanging two rows of an array changes the sign of its determinant. (Hint add a row to another, subtract the other way and add back the first way; or something like that)
And what good is all this?
