In
mathematics, the determinant of a square matrix is a value computed from the elements of the matrix by certain,
equivalent rules. The determinant provides important information when the
matrix consists of the coefficients of a system of linear equations, and
when it describes a linear transformation: in the first case the system has a unique
solution if and only if the determinant
is nonzero; in the second case that same condition means that the
transformation has an inverse operation. A geometrical interpretation can be given to
the value of the determinant of a square matrix with real entries: the absolute value of the determinant is the scale factor by which area or volume is
multiplied under the associated linear transformation, while its sign indicates
whether the transformation preserves orientation. Thus a
2 × 2 matrix with determinant −2, when applied to a region of
the plane with finite area, will transform that region into one with twice the
area, while reversing its orientation.
The
determinant of a matrix A is denoted det (A), det A, or |A|. In the case where the matrix
entries are written out in full, the determinant is denoted by surrounding the
matrix entries by vertical bars instead of the brackets or parentheses of the
matrix. For
instance, the determinant of the matrix
is written and has the value
(Our solved example in mathguru.com uses
this concept)
Properties
characterizing the determinant
The
determinant has the following properties:
1. If A is a triangular matrix, i.e. a_{i}_{,j} = 0 whenever i > j or, alternatively, whenever i < j, then
the product of the diagonal entries
of A. For example, the determinant of the identity matrix
is one.
2. If B results from A by
interchanging two rows or two columns, then det(B) =
−det(A). The determinant is called alternating (as
a function of the rows or
columns of the matrix).
3. If B results from A by
multiplying one row or column with a number c,then det
(B)= c .
det(A). As a consequence, multiplying the whole matrix by c yields
4. If B results from A by
adding a multiple of one row to another row, or a multiple of one column to
another column, then
(Our solved
example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Determinant
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