MATH 323 Lecture 7

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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:

Theorem 2.1.1

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:

Theorem 2.1.2

If is a matrix, then

Properties of Determinants

Lemma 2.2.1

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:

  1. , where is the scalar multiple used to produce

In all cases, , and coincidentally, the determinant of for all types is:

  1. -1
  2. α
  3. 1