4.1 Definition. We will define, for each square matrix A, a scalar which is called determinant of A and denoted by det(A), or |A)|. Thus,
|
will denote the determinant of a matrix A = [aij]n×n.
The determinant is a function of the elements of the matrix, its definition can be given either directly through the elements, or inductively through determinants of submatrices of lower orders. We use the latter since it is more convenient for calcula
tions.
First we define the determinant of a matrix of order 2×2 by:
|
(1) |
In order to define determinants of arbitrary orders, it is convenient to introduce some terminology.
Assume that A is a given square matrix, and B is a square submatrix of A. The determinant of B is called a minor of the determinant of A.
Let ai,j be a fixed element of A and denote by Aij a square submatrix which is obtained by removing the i-th row and j-th column of A. The determinant of Aij, which is a minor of the determinant of A by the previous definition, is called a principal minor corresponding to aij.
The number
|
is called a cofactor of aij.
Definition 1 Given a matrix A of order n×n is the sum of elements of any row times their corresponding cofactors.
In other words,
|
(2) |
We note that in formula (2) the index i can be chosen arbitrary (1 £ i £ n), and will give the same result. Moreover, instead of fixing a row we can fix a column and the sum of elements of the column times their corresponding cofactors. Thus, for arbitrary j, 1 £ j £ n, we have
|
(3) |
Also note that Definition 1 in fact reduces the determinant of n-order to determinant of (n-1)-order. Having defined the determinant of 2-nd order by (1), we can use it to define determinants of 3-rd order, 4-th order, and so on.
Thus, determinats of third order is given by, e.g. the following formula:
|
We remark that since the determinant is the same regardless of the row or column we choose for the expansions by cofactors. Thus, it will lead to a less computation if we choose such a row or column that contains more zeros.
The method of calculation of determinants using just the definition is not convenient for matrices of higher orders, since it leads to a large colume of computations. We will see in a later section that one can use the elementary row transformations to calculate the determinants more easily. But before presenting this method we need some properties of determinants.
4.2 Properties of determinants.
Theorem 1 If a matrix has a zero row or column, then its determinant is zero.
Proof: Choose the zero row or the zero column for the expansion by cofactors.
Theorem 2 If B is a matrix which is obtained by a matrix A by interchanging any two rows or columns, then |B| = -|A|.
Proof: We postpone the proof until the end of this chapter.
Theorem 3 If two rows or two columns of a matrix A are identical, then the determinant is zero.
Proof: Assume that the i-th row and the k-row of the matrix are identical. Then the matrix B, which is obtained from the matrix A by interchanging i-th row and j-th row, is identical with A. On the other hand, Theorem 2 implies that |B| = -|A|. Thus, |A| = -|A|, hence |A| = 0.
Theorem 4 If a matrix B is obtained from a matrix A by multiplying a row or a column by a scalar l, then |B| = l|A|.
Proof: Assume that the i-th row of B is obtained from the i-row of A by multiplying by l - thus it has elements lai1,lai2,...,lain. Let Ai1,...,Ain be the principal minors corresponding to ai1,...ain. Then we have, by formula (2),
|
Theorem 5 If A is a matrix of order n×n and l is an arbitrary scalar, then |lA| = ln|A|.
Proof We apply Theorem 4 n-times to |lA| as follows:
|
Theorem 6 If B is obtained from A by adding to a row (a column) a scalar times another row (resp., another column), then |B| = |A|.
Proof: Let B be obtained from A by adding to the i-th row a times the k-th row. Write the expansion of |B| according to formula (2), by cofactors corresponding to the i-th row. Since the elements of the i-th row of B are ai1+aak1,..., ain+aakn and the corresponding cofactors of B
and of A are the same, we have
|
The first term in the right hand side of (4) is equal |A|, while the second is equall to a times the determinant of a matrix which has the k-th row equal to the i-th row. By Theorem 3, this second term is zero. Therefore, |B| = |A|.
Theorem 7 |At| = |A|.
Proof: This is because the expansion of |At| by cofactors corresponding to the i-th row is identical to the expansion of |A| by cofactors corresponding to the i-th column.
Theorem 8 The determinant of a upper or lower triangular matrix is equal to the product of its diagonal elements.
Proof: Let A be a lower triangular matrix. According to the expansion (2) by cofactors of the elements of the first row, we have
|
Since A22 is again a lower triangular matrix, we can apply the same argument to get
|
Finally, we have the follwoing important theorem.
Theorem 9 |AB| = |A||B| for any two matrices (of the same order n×n).
4.3 Calculation of determinats by reducing to a triangular form.
One can use the definition of determinant (formula (?)) to calculate determinants of 3-rd order, but if a matrix has order 4×4 or higher, then it is not convenient to use formula (2) because it involves a huge volume of computations. In this section, we show that there is an alternative and more efficient method for calculating determinants. This method relies on the reduction of a matrix to an upper triangular form using elementary row or column transformations.
Recall that, for a given matrix A, we can use the elementary row transformations of type (i)-(iii) to reduce it to a row reduced form. Each such elementary row transformation is a result of multiplication (from the left) of the matrix A by an elementary matrix of a type (i)-(iii). In order to reduce A to an upper triangular form without the requirement that the diagonal elements must be 1 or 0, we can restrict ourselves to the use of elementary matrices of type (i) (interchanging any two rows of the identity matrix) and type (iii) (add to a row of the identity matrix a scalar times another row). The determinants of the corresponding elementary matrices are -1 and 1, respectively. Thus, there are elementary matrices E1,...,Ek such that
|
is an upper triangular matrix. By Theorem (?), we have
|
(7) |
Thus, the calculation of |A| reduced to the calculation of |Au, which is just the product of the diagonal elements.
As a corollary of (?) we immediately obtain the following fact:
A matrix A is invertible if and only if |A| ¹ 0.
In fact, A is invertible if and only if it its row reduced form does not contain any zero rows, i.e. there is a decomposition of the form (?) in which the matrix Au has all diagonal elements different from 0. This is exactly the case when |Au| ¹ 0, or |A| ¹ 0.
Examples:
4.4 Formula for inverse matrices:
There is a formula for the inverse matrices in terms of determinants and cofactors. Although this formula is less efficient for computing the inverse matrices, it is a succinct and important formula (from the point of view of the theory) which deserves a place here.
We denote the elements of the inverse matrix A-1 by aij(-1). Then
|
(8) |
Proof: The proof is obtained in a straighforward manner if we multiply the matrix A with the matrix, whose elements are given by formula (?). Thus, denoting by B = [aij(-1) the matrix elements of which are given by (?), and
AB = C = [cij], we have
|
We see that if i ¹ j, then the sum in the right hand side of formula (?) is precisely the determinant of a matrix which is obtained from the matrix A by replacing the j-th row by the i-row. Such a matrix has two identical rows, so the determinant is zero according to Theorem (?). Thus, cij = 0 if i = j.
If i = j, then the sum in the right hand side of (?) is precisely the determinant of A, which implies cii = 1. Thus, C = I, so that A(-1) = B.
Example:
4.5 Cramer's formula.
As we have seen, if A is an invertible square matrix then for every b Î Cn, equation
|
(10) |
has a unique solution. In this case there is a formula for the solution of (1) in terms of determinants of matrices constructed from (?), which is called Cramer's formula. As the preceding formula for the inverse matrices, Cramer's formula is not always efficient in solving the equations, but it is a succinct and important formula deserving attention.
Let us denote by Bi a matrix which is obtained from A by replacing the i-th column by the column b. Then the solution xi is found by the following formula:
|
(11) |
Proof: Assume that xi, i = 1,2,...,n. is the solution of equation (?). We show that the formula (?) holds. By Theorem (?), we know that the determinant is not changed if we add to a column some linear combination of other columns. In the next formula, we add to the first column of the determinant x2 times second column, x3 times third column, and so on. Thus, we have
|