-
A determinant on
is an alternating multilinear function such that . We denote this as or . We can get an explicit, recursive formula for the determinant below.
-
(Friedberg 4.5.1) Let
be an alternating -linear function on . For each matrix and each , we can define the determinant as Where
is the matrix obtained from by deleting the -th row and -th column. -
(Friedberg 4.5.2) There exists a determinant on
for any positive integer . -
(Friedberg 4.6) Any determinant
on has the following properties -
If
is a matrix obtained from by multiplying each entry of some row of by a scalar , then - (Friedberg e4.3.6) If
then
- (Friedberg e4.3.6) If
-
If two rows of
are identical, then -
If
is a matrix obtained from by interchanging two rows, then -
If one row of
consists entirely of zero entries, then -
If
is a matrix obtained from by adding a multiple of row to row , , then
-
-
(Friedberg 4.6.1) Let
be elementary matrices in of types respectively. If is obtained by multiplying a row of by nonzero scalar then for any determinant -
(Friedberg 4.7) Let
be a determinant on and let such that . (i.e., the matrix is not invertible). Then -
(Friedberg Lem.4.8) If
is an elementary matrix with entries from and is a determinant on , then For any
-
(Friedberg 4.8) Let
be a determinant on and let . Then -
(Friedberg 4.8.1) Let
be invertible. Then -
(Friedberg 4.8.2) Let
. The following are equivalent is not invertible
-
(Friedberg 4.9) There is exactly one determinant on
-
(Friedberg 4.10) For any
-
(Friedberg 4.10.1) Any statement about determinants involving rows can be restated in terms of the columns and vice versa.
-
(Friedberg 4.11) The determinant of an upper triangular square matrix is the product of its diagonal entries.
-
(Friedberg 4.11.1) If
is a lower triangular matrix, then the determinant of the matrix is the product of its diagonal entries. -
(Friedberg e4.3.8) If
is skew symmetric and is odd, then -
(Friedberg 4.12) Cramer’s Rule Let
be the matrix form of a system of linear equations with equations and unknowns. Let and . If then this system has a unique solution. Also
then Where
is the matrix obtained from by replacing the -th column with . -
A system of linear equations with integral coefficients has an integral solution if
. -
(Friedberg e4.3.9) Let
be a matrix written in the form Where
are square matrices. Then -
(Friedberg e4.3.15) If
then -
The results above extend to linear transformations, which are isomorphic to matricesSimilarly, (Friedberg 5.5) if
and are bases for linear operator , then We call the above the determinant of the linear operator
defined as - (Friedberg 5.6) The following hold for linear operator
is invertible if and only if - If
is invertible, then - If
is linear, then - If
is any scalar and any basis for , then Where
- (Friedberg 5.6) The following hold for linear operator