In this section, we define the determinant, and we present one way to compute it. Then we discuss some of the many wonderful properties the determinant enjoys.
Subsection4.1.1The Definition of the Determinant
The determinant of a square matrix is a real number It is defined via its behavior with respect to row operations; this means we can use row reduction to compute it. We will give a recursive formula for the determinant in Section 4.2. We will also show in this subsection that the determinant is related to invertibility, and in Section 4.3 that it is related to volumes.
Definition
The determinant is a function
satisfying the following properties:
Doing a row replacement on does not change
Scaling a row of by a scalar multiplies the determinant by
Swapping two rows of a matrix multiplies the determinant by
The determinant of the identity matrix is equal to
In other words, to every square matrix we assign a number in a way that satisfies the above properties.
In each of the first three cases, doing a row operation on a matrix scales the determinant by a nonzero number. (Multiplying a row by zero is not a row operation.) Therefore, doing row operations on a square matrix does not change whether or not the determinant is zero.
The main motivation behind using these particular defining properties is geometric: see Section 4.3. Another motivation for this definition is that it tells us how to compute the determinant: we row reduce and keep track of the changes.
Example
Let us compute First we row reduce, then we compute the determinant in the opposite order:
The reduced row echelon form of the matrix is the identity matrix so its determinant is The second-last step in the row reduction was a row replacement, so the second-final matrix also has determinant The previous step in the row reduction was a row scaling by since (the determinant of the second matrix times ) is the determinant of the second matrix must be The first step in the row reduction was a row swap, so the determinant of the first matrix is negative the determinant of the second. Thus, the determinant of the original matrix is
Note that our answer agrees with this definition of the determinant.
Here is the general method for computing determinants using row reduction.
Recipe: Computing determinants by row reducing
Let be a square matrix. Suppose that you do some number of row operations on to obtain a matrix in row echelon form. Then
where is the number of row swaps performed.
In other words, the determinant of is the product of diagonal entries of the row echelon form times a factor of coming from the number of row swaps you made, divided by the product of the scaling factors used in the row reduction.
This is an efficient way of computing the determinant of a large matrix, either by hand or by computer. The computational complexity of row reduction is by contrast, the cofactor expansion algorithm we will learn in Section 4.2 has complexity which is much larger. (Cofactor expansion has other uses.)
If a matrix is already in row echelon form, then you can simply read off the determinant as the product of the diagonal entries. It turns out this is true for a slightly larger class of matrices called triangular.
Definition
The diagonal entries of a matrix are the entries
A square matrix is called upper-triangular if its nonzero entries all lie above the diagonal, and it is called lower-triangular if its nonzero entries all lie below the diagonal. It is called diagonal if all of its nonzero entries lie on the diagonal, i.e., if it is both upper-triangular and lower-triangular.
Proposition
Let be an matrix.
If has a zero row or column, then
If is upper-triangular or lower-triangular, then is the product of its diagonal entries.
Proof
Suppose that has a zero row. Let be the matrix obtained by negating the zero row. Then by the second defining property. But so
Putting these together yields so
Now suppose that has a zero column. Then is not invertible by the invertible matrix theorem in Section 3.6, so its reduced row echelon form has a zero row. Since row operations do not change whether the determinant is zero, we conclude
First suppose that is upper-triangular, and that one of the diagonal entries is zero, say We can perform row operations to clear the entries above the nonzero diagonal entries:
In the resulting matrix, the th row is zero, so by the first part.
Still assuming that is upper-triangular, now suppose that all of the diagonal entries of are nonzero. Then can be transformed to the identity matrix by scaling the diagonal entries and then doing row replacements:
Since and we scaled by the reciprocals of the diagonal entries, this implies is the product of the diagonal entries.
The same argument works for lower triangular matrices, except that the the row replacements go down instead of up.
A matrix can always be transformed into row echelon form by a series of row operations, and a matrix in row echelon form is upper-triangular. Therefore, we have completely justified the recipe for computing the determinant.
The determinant is characterized by its defining properties, since we can compute the determinant of any matrix using row reduction, as in the above recipe. However, we have not yet proved the existence of a function satisfying the defining properties! Row reducing will compute the determinant if it exists, but we cannot use row reduction to prove existence, because we do not yet know that you compute the same number by row reducing in two different ways.
Theorem(Existence of the determinant)
There exists one and only one function from the set of square matrices to the real numbers, that satisfies the four defining properties.
We will prove the existence theorem in Section 4.2, by exhibiting a recursive formula for the determinant. Again, the real content of the existence theorem is:
No matter which row operations you do, you will always compute the same value for the determinant.
Subsection4.1.2Magical Properties of the Determinant¶ permalink
If is invertible, then it has a pivot in every row and column by the invertible matrix theorem in Section 3.6, so its reduced row echelon form is the identity matrix. Since row operations do not change whether the determinant is zero, and since this implies Conversely, if is not invertible, then it is row equivalent to a matrix with a zero row. Again, row operations do not change whether the determinant is nonzero, so in this case
If the columns of are linearly dependent, then is not invertible by condition 4 of the invertible matrix theorem in Section 3.6. Suppose now that the rows of are linearly dependent. If are the rows of then one of the rows is in the span of the others, so we have an equation like
If we perform the following row operations on
then the second row of the resulting matrix is zero. Hence is not invertible in this case either.
In particular, if two rows/columns of are multiples of each other, then We also recover the fact that a matrix with a row or column of zeros has determinant zero.
In this proof, we need to use the notion of an elementary matrix. This is a matrix obtained by doing one row operation to the identity matrix. There are three kinds of elementary matrices: those arising from row replacement, row scaling, and row swaps:
The important property of elementary matrices is the following claim.
Claim: If is the elementary matrix for a row operation, then is the matrix obtained by performing the same row operation on
In other words, left-multiplication by an elementary matrix applies a row operation. For example,
The proof of the Claim is by direct calculation; we leave it to the reader to generalize the above equalities to matrices.
As a consequence of the Claim and the four defining properties, we have the following observation. Let be any square matrix.
If is the elementary matrix for a row replacement, then In other words, left-multiplication by does not change the determinant.
If is the elementary matrix for a row scale by a factor of then In other words, left-multiplication by scales the determinant by a factor of
If is the elementary matrix for a row swap, then In other words, left-multiplication by negates the determinant.
Now we turn to the proof of the multiplicativity property. Suppose to begin that is not invertible. Then is also not invertible: otherwise, implies By the invertibility property, both sides of the equation are zero.
Now assume that is invertible, so Define a function
We claim that satisfies the four defining properties of the determinant.
Let be the matrix obtained by doing a row replacement on and let be the elementary matrix for this row replacement, so Since left-multiplication by does not change the determinant, we have so
Let be the matrix obtained by scaling a row of by a factor of and let be the elementary matrix for this row replacement, so Since left-multiplication by scales the determinant by a factor of we have so
Let be the matrix obtained by swapping two rows of and let be the elementary matrix for this row replacement, so Since left-multiplication by negates the determinant, we have so
We have
Since satisfies the four defining properties of the determinant, it is equal to the determinant by the existence theorem. In other words, for all matrices we have
Multiplying through by gives
Recall that taking a power of a square matrix means taking products of with itself:
If is invertible, then we define
For completeness, we set if
Corollary
If is a square matrix, then
for all If is invertible, then the equation holds for all as well; in particular,
By the invertibility property, this is nonzero if and only if is invertible. On the other hand, is nonzero if and only if each which means each is invertible.
and we show that satisfies the four defining properties of the determinant. Again we use elementary matrices, also introduced in the proof of the multiplicativity property.
Let be the matrix obtained by doing a row replacement on and let be the elementary matrix for this row replacement, so The elementary matrix for a row replacement is either upper-triangular or lower-triangular, with ones on the diagonal:
It follows that is also either upper-triangular or lower-triangular, with ones on the diagonal, so by this proposition. By the fact and the multiplicativity property,
Let be the matrix obtained by scaling a row of by a factor of and let be the elementary matrix for this row replacement, so Then is a diagonal matrix:
Thus By the fact and the multiplicativity property,
Let be the matrix obtained by swapping two rows of and let be the elementary matrix for this row replacement, so The is equal to its own transpose:
Since (hence ) is obtained by performing one row swap on the identity matrix, we have By the fact and the multiplicativity property,
Since we have
Since satisfies the four defining properties of the determinant, it is equal to the determinant by the existence theorem. In other words, for all matrices we have
The transpose property is very useful. For concreteness, we note that means, for instance, that
This implies that the determinant has the curious feature that it also behaves well with respect to column operations. Indeed, a column operation on is the same as a row operation on and
Corollary
The determinant satisfies the following properties with respect to column operations:
Doing a column replacement on does not change
Scaling a column of by a scalar multiplies the determinant by
Swapping two columns of a matrix multiplies the determinant by
The previous corollary makes it easier to compute the determinant: one is allowed to do row and column operations when simplifying the matrix. (Of course, one still has to keep track of how the row and column operations change the determinant.)
By the first defining property, scaling any row of a matrix by a number scales the determinant by a factor of This implies that satisfies the second property, i.e., that
We claim that If is in then
for some scalars Let be the matrix with rows so By performing the row operations
the first row of the matrix becomes
Therefore,
Doing the opposite row operations
to the matrix with rows shows that
which finishes the proof of the first property in this case.
Now suppose that is not in This implies that is linearly dependent (otherwise it would form a basis for ), so = 0. If is not in then is linearly dependent by the increasing span criterion in Section 2.5, so for all as the matrix with rows is not invertible. Hence we may assume is in By the above argument with the roles of and reversed, we have
For we note that
By the previously handled case, we know that is linear:
Multiplying both sides by we see that is linear.
For example, we have
By the transpose property, the determinant is also multilinear in the columns of a matrix:
In more theoretical treatments of the topic, where row reduction plays a secondary role, the defining properties of the determinant are often taken to be:
The determinant is multilinear in the rows of
If has two identical rows, then
The determinant of the identity matrix is equal to one.
We have already shown that our four defining properties imply these three. Conversely, we will prove that these three alternative properties imply our four, so that both sets of properties are equivalent.
Defining property is just the second defining property in Section 3.3. Suppose that the rows of are If we perform the row replacement on then the rows of our new matrix are so by linearity in the th row,
where because is repeated. Thus, the alternative defining properties imply our first two defining properties. For the third, suppose that we want to swap row with row Using the second alternative defining property and multilinearity in the th and th rows, we have
