We define the determinant of a square matrix in terms of cofactor expansion along the first row.
DET-0010: Definition of the Determinant – Expansion Along the First Row
In this module we will define a function that assigns to each square matrix a scalar output called the determinant of . For matrices with real number entries, the outputs of the determinant function will be real numbers. We will denote the determinant of by .
One important property of the determinant is its connection to matrix inverses. We will find that a matrix is singular if and only if . For nonsingular matrices, we will establish a formula that gives the inverse of a matrix exclusively in terms of determinants.
Determinants of Square Matrices ()
To start from the beginning, let us define the determinant of a matrix.
It is important to note that this definition is consistent with our goal of making a connection between determinants and invertibility. Observe that exists if and only if .Now we proceed to matrices. According to Formula form:detinverse of MAT-0050, the inverse of a nonsingular matrix is given by Observe that exists if and only if . We will call the number the determinant of .
Note the distinction between the square bracket notation associated with the matrix and the vertical bar notation used to denote the determinant in expression (eq:twobytwodet).Our next goal is to define the determinant for a matrix. Let
Our definition will require three minor matrices associated with .
- is obtained from by deleting the first row and the first column of
.
- is obtained from by deleting the first row and the second column of
.
- is obtained from by deleting the first row and the third column of
.
We are now ready to define the determinant of a matrix.
We would like to point out several important features of this definition:
- The coefficients , and in front of determinants of minor matrices are the entries of the first row of matrix .
- To remember which minor matrix is associated with each first row
entry, cross out the row and column that the entry is in. For example,
minor matrix is found by crossing out the row and column that is
in.
- Coefficients follow an alternating sign pattern: , , .
Definition of the Determinant
Let be an matrix. Define to be an matrix obtained from by deleting the first row and the column of . We say that is the -minor of .
Recall that for a matrix we have We want to follow the same pattern to define the determinant of a larger matrix. A distinct feature of this expression is the alternating sign pattern. We want to preserve this feature as we increase matrix size. Before we present a formal definition, let’s take a look at an example.
Now we are ready to give the general formula. Pay close attention to how we will use subscripts to capture the alternating sign pattern.
Naturally, we would like to condense this formula. To accomplish this, let We will refer to as the -cofactor of . When we use the cofactor notation, the expression in Definition def:toprowexpansion turns into the following:The process of computing the determinant given by Definition def:toprowexpansion is called the cofactor expansion along the first row. We will later show that we can expand along any row or column of a matrix and obtain the same value. This surprising result, known as the Laplace Expansion Theorem, will be the subject of DET-0050.
We conclude this section with a simple, but useful lemma.
- Proof
- Left to the reader.
Practice Problems
Show that Definition def:toprowexpansion is consistent with Definition def:twobytwodet by verifying that both produce
the same result when applied to a matrix.
Let be a matrix obtained from of Problem prob:2x2det2 by switching the first and the second
row of . Compute the determinant of . What do you observe?
Make a conjecture about what happens to the determinant of a matrix if two rows of
a matrix are switched. Prove your conjecture for a matrix.
Let be a matrix obtained from of Problem prob:2x2det2 by multiplying the middle row by .
Compute the determinant of . What do you observe?
Make a conjecture about what happens to the determinant of a matrix if
one of the rows is multiplied by a constant. Prove your conjecture for a
matrix.
Let be a matrix obtained from of Problem prob:2x2det2 by multiplying by . Compute the
determinant of . What do you observe?
Make a conjecture about what happens to the determinant of a matrix if the matrix
is multiplied by a constant. Prove your conjecture for a matrix.
Let be a matrix obtained from of Problem prob:2x2det2 by adding twice the third row to the
first. Compute the determinant of . What do you observe?