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Example:Evaluate using Properties of Determinants

 Post to:   Explanation:

Determinant

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. ai,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