Linear Algebra/Properties of Determinants

As described above, we want a formula to determine whether an $$n \! \times \! n$$ matrix is nonsingular. We will not begin by stating such a formula. Instead, we will begin by considering the function that such a formula calculates. We will define the function by its properties, then prove that the function with these properties exists and is unique and also describe formulas that compute this function. (Because we will show that the function exists and is unique, from the start we will say "$$ \det(T) $$" instead of "if there is a determinant function then $$ \det(T) $$" and "the determinant" instead of "any determinant".)

The first result shows that a function satisfying these conditions gives a criteria for nonsingularity. (Its last sentence is that, in the context of the first three conditions, (4) is equivalent to the condition that the determinant of an echelon form matrix is the product down the diagonal.)

That result gives us a way to compute the value of a determinant function on a matrix. Do Gaussian reduction, keeping track of any changes of sign caused by row swaps and any scalars that are factored out, and then finish by multiplying down the diagonal of the echelon form result. This procedure takes the same time as Gauss' method and so is sufficiently fast to be practical on the size matrices that we see in this book.

The prior example illustrates an important point. Although we have not yet found a $$4 \! \times \! 4$$ determinant formula, if one exists then we know what value it gives to the matrix &mdash; if there is a function with properties (1)-(4) then on the above matrix the function must return $$5$$.

The "if there is an $$n \! \times \! n$$ determinant function" emphasizes that, although we can use Gauss' method to compute the only value that a determinant function could possibly return, we haven't yet shown that such a determinant function exists for all $$n$$. In the rest of the section we will produce determinant functions.

Exercises
''For these, assume that an $$n \! \times \! n$$ determinant function exists for all $$n$$.''

/Solutions/