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A matrix is nonsingular iff its determinant is nonzero.
Let be a matrix.
Let be a matrix obtained from by deletion of the row and column containing . The matrices are called minors of .
Let . This is called the cofactor of
The determinant of a matrix is a scalar associated with that is defined inductively as follows:
The determinant can be expressed as a cofactor expansion using any row or column of :
It is beneficial to choose a row or column with the maximum 0 entries. This process is called inspection
Determinant of 2 × 2 Matrix
For a matrix , the minors are:
and the determinant could be defined as:
The corresponding signs for each entry are defined by the following matrix:
If is a matrix, then
Properties of Determinants
Given a matrix , let be the cofactor of for .
First case is trivial: definition of determinant
Second case: Let be the matrix formed by replacing the th row of with the th row of .
because the th row is deleted, but by MATH 323 Theorems#Theorem 2.1.4, the determinant of a matrix with two identical rows is 0.
Multiplication by Elementary Matrices
What happens when we take the determinant of the product of a matrix and an elementary matrix ?
If is of type:
- , where is the scalar multiple used to produce
In all cases, , and coincidentally, the determinant of for all types is: