Finding the Determinant

In this section we will define a function that assigns to each square matrix a scalar output called the determinant of . We will denote the determinant of by . For a matrix with real number entries, the output of the determinant function will always be a real number.

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. This property will be addressed in detail in Poperties of the Determinant, and Determinants and Inverses of Nonsingular Matrices.

Geometrically speaking, the determinant of a matrix of a linear transformation is the factor by which the area (or volume or hypervolume) is scaled by the transformation. This will be discussed in Determinants as Areas and Volumes.

Cofactor Expansion Along the Top Row

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, 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).

The easiest way to understand the definition of the determinant for a matrix is to start with an example.

We now formalize what we learned in Example ex:threebythreedet1.

We will now reiterate several important features of this definition and introduce some vocabulary:

  • The coefficients , and are the entries of the first row of matrix . Coefficients in the formula follow an alternating sign pattern: , , . This pattern will persist in the determinant formulas for determinants of larger matrices.
  • When using equation (eq:det3by3), we compute determinants of three matrices: These matrices are called minor matrices. To form each minor matrix, cross out the row and column that the corresponding coefficient is in. For example, the minor matrix corresponding to coefficient is found by crossing out the row and column that is in.

  • The process for finding the determinant described in Definition def:threebythreedet is referred to as a cofactor expansion along the top row.

We are starting to observe a certain pattern in the process of computing the determinant. This pattern will persist for larger matrices. Let’s take a look at a matrix.

Cofactor Expansion Along the First Column

We defined the determinant of a matrix in terms of cofactor expansion along the top row. We will now see what happens when we expand along the first column instead. We will refer to this process as cofactor expansion along the first column. Surprisingly, both expansions yield the same result. To illustrate this, let’s revisit Examples ex:threebythreedet1 and ex:expansiontoprow.

Let’s go through this process again for a larger matrix.

Let In Example ex:expansiontoprow we found that . We will now try to expand along the fist column.

When computing determinants of the four matrices below, try different approaches. You might want to expand along the first row for some of them, and along the first column for others. Looking for where zeros are located will help you decide what to try.

In Example init:expansionfirstcol1 and Exploration init:expansionfirstcol2 we were careful not to claim at the outset that we were finding the determinant of the matrix by cofactor expansion along the first column; we merely observed that the resulting value was equal to the determinant. It is possible to prove that both expansions produce the same result (Theorem th:rowcolexpequivalence). Therefore the determinant of a matrix can be defined in terms of cofactor expansion along the first row or column.

Cofactor Expansion Along Any Row or Column

We originally defined the determinant of a matrix via expansion along the top row of the matrix. We later observed that expansion along the first column produces the same result. It turns out that the value of the determinant can be computed by expanding along any row or column. This result is known as the Laplace Expansion Theorem (th:laplace1).

When expanding along an arbitrary row or column, we will continue to follow the two patterns we observed earlier.

  • The alternating sign pattern for coefficients will follow the checkerboard pattern below.
  • Minor matrices will be formed by eliminating the row and column that the corresponding coefficient is in.

To illustrate this, let’s take another look at matrix from Example ex:expansiontoprow.

It is clear that having zeros as entries in the matrix reduces the number of computations necessary to find the determinant. The following example demonstrates how to use zeros to our advantage.

A Note on Equivalency

We initially introduced the determinant of a matrix via cofactor expansion along the top row. We later observed that cofactor expansion along any row or column produces the same result. We have to be careful, however, not to use a few examples as “proof” that all cofactor expansions are equivalent. Such claims need to be carefully supported with general proofs. Unfortunately, in this case, the proofs are tedious and conceptually unenlightening. An interested reader can find them in Tedious Proofs Concerning Determinants.

Determinants of Some Special Matrices

We know that we can find the determinant of a matrix by cofactor expansion along the top row or the first column. (See Theorem th:rowcolexpequivalence of Tedious Proofs Concerning Determinants for proof.) This property gives rise to a useful result.

Proof
See Practice Problem prob:detOfTrans.

As we observed earlier, having zeros in a matrix makes it easier for us to compute its determinant. Recall that that a square matrix is upper-triangular if all of the entries below the main diagonal are zero. Similarly, a square matrix is called lower-triangular if all of the entries above the main diagonal are zero. Together, upper and lower triangular matrices are categorized as triangular matrices.

Proof
We proceed by induction on , where is an matrix. It is easy to see that this result holds for . Suppose that the result holds for triangular matrices. We need to show that it holds for triangular matrices.

Suppose is a triangular matrix. Then, with the exception of , the entries in the first row (or column) of are guaranteed to be zeros. We will take advantage of these zeros and expand along the first row (or column) of . As we do so, we obtain a single product of and the determinant of a minor matrix obtained by crossing out the first row and column of . But this minor matrix is also a triangular matrix with diagonal etries . By induction hypothesis, its determinant is equal to the product of its diagonal entries, . Therefore This completes the proof.

As an immediate consequence of this theorem, we have the following result.

We first introduced block matrices in Block Matrix Multiplication. Matrices of the form and , where , are square matrices and is the zero matrix, are said to be block triangular. The following theorem makes it easy to compute determinants of such matrices.

Proof
Write and proceed by induction on , where is . If , then the result follows from cofactor expansion along the first column. In general, let denote the matrix obtained from by deleting row and column 1. Then the cofactor expansion along the first column is where are the entries in the first column of . Observe that where , denotes matrix with column 1 and row deleted, and denotes matrix with with row deleted.

Since is a matrix, by the induction hypothesis, This gives us

The lower triangular case is similar.

Practice Problems

Problems prob:2x2det1-prob:laplace Find the determinant of each matrix.

Answer:
Answer:
Answer:
Show that Definition def:toprowexpansion is consistent with Definition def:twobytwodet by verifying that both produce the same result when applied to a matrix.
Prove Theorem th:detoftrans.
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?
Make a conjecture about what happens to the determinant of a matrix if a multiple of one row is added to another row. Prove your conjecture for a matrix.
Is it true that ?

Text Source

The text in this section partially comes from Section 3.1 of Keith Nicholson’s Linear Algebra with Applications (CC-BY-NC-SA).