You already know the definition of the determinant of a 2 by 2 matrix
[ a | b ] |
[ c | d ] |
The determinant of this matrix is ad-bc. You also know how to calculate the determinant of a 3 by 3 matrix (remember the cross product!). Before we define the determinant of an arbitrary n by n matrix, consider an example. Let Maple find the determinant of the following 4 by 4 matrix:
A := | [ a_{11} | a_{12} | a_{13} | a_{14} ] |
[ a_{21} | a_{22} | a_{23} | a_{24} ] | |
[ a_{31} | a_{32} | a_{33} | a_{34} ] | |
[ a_{41} | a_{42} | a_{43} | a_{44} ] |
det(A) = a_{11} a_{22} a_{33} a_{44} - a_{11} a_{22} a_{43} a_{34} - a_{11} a_{32} a_{23} a_{44} + a_{11} a_{32} a_{43} a_{24} + a_{11} a_{42} a_{23} a_{34} - a_{11} a_{42} a_{33} a_{24} - a_{21} a_{12} a_{33} a_{44} + a_{21} a_{12} a_{43} a_{34} + a_{21} a_{32} a_{13} a_{44} - a_{21} a_{32} a_{43} a_{14} - a_{21} a_{42} a_{13} a_{34} + a_{21} a_{42} a_{33} a_{14} + a_{31} a_{12} a_{23} a_{44} - a_{31} a_{12} a_{43} a_{24} - a_{31} a_{22} a_{13} a_{44} + a_{31} a_{22} a_{43} a_{14} + a_{31} a_{42} a_{13} a_{24} - a_{31} a_{42} a_{23} a_{14} - a_{41} a_{12} a_{23} a_{34} + a_{41} a_{12} a_{33} a_{24} + a_{41} a_{22} a_{13} a_{34} - a_{41} a_{22} a_{33} a_{14} - a_{41} a_{32} a_{13} a_{24} + a_{41} a_{32} a_{23} a_{14}
One can notice the following features of this expression:
has the negative sign.
has positive sign, we can say that the sign of a term is positive if the permutation corresponding to this term is obtained from the trivial permutation by an even number of transpositions, otherwise the sign is negative.
Thus we can give the
following definition of the determinant of a square matrix A.
Let A be a square n by n matrix. List all permutations p=(p_{1},...,p_{n})
of numbers {1,...,n} -- there are n factorial of them. For each permutation
p consider the product
of entries of matrix A. Here (-1)^{p} denotes the sign of the permutation p that is "+" if p can be obtained from the trivial permutation by an even number of transpositions and "-" otherwise. Then the determinant det(A) is equal to the sum of all these (n factorial) products.
The sign of a permutation can be calculated in a different way. We say
that a permutation p=(p_{1},...,p_{n}) of numbers {1,...,n} has an
inversion if a larger number precedes a smaller number in
this permutation. For example, permutation (4,3,2,1) has 6 inversions:
4 precedes 3, 2 and 1 (three inversions), 3 precedes 2 and 1 (two inversions),
2 precedes 1 (one inversion).
Theorem. The sign of a permutation is "+"
if and only if the number of inversions in this permutation is even.
To see a hint to a prove of this theorem, click here.
Thus we have two equivalent definitions of the sign of a permutation. We call a permutation even if its sign is "+", otherwise the permutation is called odd.
Here we establish several important properties of determinants. We start with some simple properties and then deduce more and more complex ones. Each step will be relatively simple but at the end we shall get some very non-trivial statements.
Theorem. Let A be an n by n matrix. Then the following conditions hold.
Theorem. Let A be an n by n matrix. Then the following conditions hold.
where C_{nn} is the cofactor of entry A(n,n) that is the determinant of the matrix obtained by deleting the last row and the last column of matrix A.
Theorem. Let A and B be n by n matrices. Then the following conditions hold.
a) If A is an elementary matrix then det(AB)=det(A)det(B).
b) A is invertible if and only if det(A) is not 0.
c) det(AB)=det(A)det(B) (the matrix A is arbitrary here).
Proof. Indeed, if AB=I the by the third theorem about determinants
Therefore det(A) is not zero (otherwise 0=1). Again by the third theorem about determinants this implies that A is invertible.
Corollary 2.If A is an n by n matrix and i and j are different numbers from 1 to n then
Proof. Indeed, let us replace the i-th row of A by the j-th row of A and call the resulting matrix B. Then the sum in the left hand side of our equality is the cofactor expansion of the matrix B along the j-th row. Thus by the second theorem about determinants this sum is equal to det(B). But B has two equal rows (row i and row j) thus det(B)=0, so the equality holds.
Definition. If A is any n by n matrix and C_{ij} is the cofactor of A(i,j) then the matrix
[ C_{11} | C_{21} | ... | C_{n1} ] |
[ C_{12} | C_{22} | ... | C_{n2} ] |
............................ | |||
[ C_{1n} | C_{2n} | ... | C_{nn} ] |
Corollary 3. If A is an n by n matrix then
Proof. We shall prove only the first equality because the
second one is similar. Let B=A*adj(A). Then B(i,j) is the
product of the i-th row of A by
the j-th column of adj(A). But if we multiply this row and this column,
we get the sum from
the previous corollary. As we noticed before, if i equals j then this sum is
the cofactor expansion of A along the i-th
row, so it is equal to det(A). If i is not equal to j then the sum is 0.
Thus
Corollary 4. If A is invertible then
Proof. Indeed, if A is invertible then by the third theorem about determinants det(A) is not zero. Therefore we can multiply the equality from the previous corollary by 1/det(A) and get
[ det(A)*x_{1} ] [ b_{1}C_{11}+b_{2}C_{21}+...+b_{n}C_{n1} ] [ det(A)*x_{2} ] [ b_{1}C_{12}+b_{2}C_{22}+...+b_{n}C_{n2} ] ....... = ................ [ det(A)*x_{n} ] [ b_{1}C_{1n}+b_{2}C_{2n}+...+b_{n}C_{nn} ]From this, we can deduce that for every i=1,2,...,n
(Cramer's rule) If A is an invertible matrix then the solution of the system of linear equalitions
can be written as
where A_{i} is the matrix obtained from A by replacing the i-th column by
the column vector b.
Notice that Cramer's rule is much less efficient for large systems
of equations than the Gauss-Jordan algorithm. But it is more useful is
research applications when we want to find out general properties of
systems of linear equations. Notice also that a form of Cramer's rule
can be used even if det(A)=0, that is when A is not invertible. We shall
return to it later.