In this lecture we derive several useful properties of the determinant.
In order to fully understand this lecture you need to remember the main results derived in the lecture on the determinant of an elementary matrix.
Table of contents
The first result concerns the determinant of a triangular matrix.
Proposition Let be a triangular matrix (either upper or lower). Then, the determinant of is equal to the product of its diagonal entries:
Suppose that is lower triangular. Denote by the set of all permutations of the first natural numbers. Let be the permutation in which the numbers are sorted in increasing order. The parity of is even and its sign is because it does not contain any inversion (see the lecture on the sign of a permutation). Then, the determinant of iswhere in step we have used the fact that for all permutations except the productinvolves at least one entry above the main diagonal that is equal to zero. The latter fact can be proved by contradiction. Suppose the product involves only elements on the main diagonal or below it, and at least one element below it (otherwise ). Then,for all , but the inequality must be strict for at least one . Suppose that the inequality is strict for . Then, we have for . In other words, the permutation must contain different natural numbers smaller than or equal to , which is clearly impossible. This ends the proof by contradiction. Thus, we have proved the proposition for lower triangular matrices. The proof for upper triangular matrices is almost identical (we just need to reverse the inequalities in the last step).
A corollary of the proposition above follows.
Proposition Let be an identity matrix. Then,
The identity matrix is diagonal. Therefore, it is triangular and its determinant is equal to the product of its diagonal entries. The latter are all equal to . As a consequence, the determinant of is equal to .
The next proposition states an elementary but important property of the determinant.
Proposition Let be a square matrix and denote its transpose by . Then,
Denote by the set of all permutations of the first natural numbers. For any permutation , there is an inverse permutation such thatfor . If is obtained by performing a sequence of transpositions, then is obtained by performing the opposite transpositions in reverse order. Thus, the number of transpositions is the same and, as a consequence, we have that By using the concept of inverse permutation, the determinant of can be easily calculated as follows:where: in step we have used the definition of transpose; in step we have set and, as a consequence, .
The following property, while pretty intuitive, is often used to prove other properties of the determinant.
Proposition Let be a square matrix. If has a zero row (i.e., a row whose entries are all equal to zero) or a zero column, then
This property can be proved by using the definition of determinantFor every permutation , we have thatbecause the product contains one entry from each row (column), but one of the rows (columns) contains only zeros. Therefore,
We are now going to state one of the most important properties of the determinant.
Proposition Let be a square matrix. Then is invertible if and only if and it is singular if and only if
The matrix is row equivalent to a unique matrix in reduced row echelon form (RREF). Since and are row equivalent, we have thatwhere are elementary matrices. Moreover, by the properties of the determinants of elementary matrices, we have thatBut the determinant of an elementary matrix is different from zero. Therefore,where is a non-zero constant. If is invertible, is the identity matrix and If is singular, has at least one zero row because the only square RREF matrix that has no zero rows is the identity matrix, and the latter is row equivalent only to non-singular matrices. We have proved above that matrices that have a zero row have zero determinant. Thus, if is singular, andTo sum up, we have proved that all invertible matrices have non-zero determinant, and all singular matrices have zero determinant. Since a matrix is either invertible or singular, the two logical implications ("if and only if") follow.
The next proposition shows that the determinant of a product of two matrices is equal to the product of their determinants.
Proposition Let and be two matrices. Then,
If one of the two matrices is singular (i.e., not full rank), then their product is singular becauseas explained in the lecture entitled Matrix product and rank. Therefore, and at least one of or is zero, so thatThus, the statement in the proposition is true if at least one of the two matrices is singular. If neither of them is singular, then we can write them as products of elementary matrices:where and are elementary matrices. Since the determinant of a product of elementary matrices is equal to the products of their determinants, we have thatThus, we have proved that the statement in the proposition is true also in the case when the two matrices are non-singular.
The previous proposition allows to easily find the determinant of the inverse of a matrix.
Proposition Let be a invertible matrix. Then,
Sincewe have thatBut the determinant of a product equals the product of determinants:and As a consequence,Furthermore, the determinant of an invertible matrix is different from zero, so that we can divide both sides of the equation above by and obtain
This sub-section presents an easy-to-prove proposition about the multiplication of a matrix by a scalar. Before reading the proof, try to prove it by yourself as an exercise.
Proposition Let be a matrix. Then, for any scalar ,
This proposition is easily proved by using the definition of determinant.
This property is similar to the previous one.
Proposition Let be a matrix. Let be a scalar. Let be a matrix obtained from by multiplying a row (or column) by . Then,
Suppose the -th row has been multiplied by . By the definition of determinant:If instead the -th column is multiplied by , the same result holds because transposition does not change the determinant.
The determinant is linear in the rows and columns of the matrix.
Proposition Let be a matrix. Denote by the -th row of . Supposewhere and are two vectors and and are two scalars. Denote by the matrix obtained from by substituting with . Denote by the matrix obtained from by substituting with . Then,
By the definition of determinant, we have
Proposition Let be a matrix. Denote by the -th column of . Supposewhere and are two vectors and and are two scalars. Denote by the matrix obtained from by substituting with . Denote by the matrix obtained from by substituting with . Then,
This is a consequence of the previous proposition (linearity in columns) and of the fact that transpositions does not change the determinant.
One of the easiest and more convenient ways to compute the determinant of a square matrix is based on the LU decompositionwhere , and are a permutation matrix, a lower triangular and an upper triangular matrix respectively. We can writeand the determinants of , and are easy to compute:
if the number of row interchanges needed to obtain from the identity matrix is even; otherwise, ;
is equal to the product of the diagonal entries of because is lower triangular;
is equal to the product of the diagonal entries of because is upper triangular.
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