In this explainer, we will learn how to find the determinant of a triangular matrix.
Let us begin by recalling how to find the determinant of a general matrix. To do this, we need to know the definitions of minors and cofactors.
Let be a matrix of order . Then, the minor of element (denoted by ) is the determinant of the matrix obtained after removing row and column from .
Let be a matrix of order . Then, the cofactor of element (denoted by ) is where is the minor of element .
The determinant can be written using the cofactor expansion as follows.
Definition: Determinant of a 3 × 3 Matrix (Cofactor Expansion)
Let be a matrix. Then, for any fixed , or 3, the determinant of is where each is the cofactor of entry . This is known as the cofactor expansion (or the Laplace expansion) along row . Alternatively, for any fixed , or 3, we have
This is the cofactor expansion along column .
We also note that an alternate, perhaps more convenient, formulation is to write the above formulas explicitly in terms of determinants. So, for the first-row expansion, we have
An important facet of the cofactor expansion to consider is that we can choose which row or column we wish to expand along. It becomes clear how significant this is when we consider, for example, a matrix like
If we considered the first row, then the determinant calculation would be
While doable, this requires writing and calculating three determinants. On the other hand, if we used the second row instead, we would just have since and are both 0. Even though this calculation is much simpler, we still get the same result at the end because all forms of the cofactor expansion are equivalent.
Let us keep in mind the idea that choosing a row or column with more zeros in it simplifies the calculation when considering our first example.
Example 1: Finding the Determinant of a Matrix that Includes a Row of Zeros
Find the value of
To calculate the determinant of a matrix, recall that we can use the cofactor expansion along any row using the formula where , or 3, and along any column.
Although any choice of row or column will give us the same value for the determinant, it is always easier to choose one that has the greatest number of zeros. In particular, we can see that the third row is entirely zeros:
Therefore, letting , the determinant calculation will be
As the last example demonstrated, since the third row of the matrix had zeros for each entry, the determinant was zero. Naturally, since the cofactor expansion can be applied along any row or column, this same result will be true if any entire row or column of a matrix is zero, and this can be generalized to matrices of any size.
Property: Determinants with Zero Rows or Columns
If is a square matrix where each entry in a particular row or column is zero, then is zero.
Some examples of this include
Whenever we are asked to find a determinant, we should always be mindful to check whether any rows or columns are zeros, since this allows us to immediately conclude that the determinant is zero using this property.
In the next example, we will consider another special case of determinant calculation.
Example 2: Finding the Value of the Determinant of an Upper Triangular Matrix
Fill in the blank: The value of the determinant
When asked to find the determinant of a matrix, recall that we can use the cofactor expansion along any row using the formula where , or 3, and along any column.
It is always beneficial for us to choose a row or column to expand along that has the greatest number of zero entries, since this results in fewer calculations needed. If we examine the given matrix, we find that the first column and the third row are the best candidates since they both contain two entries that are zero:
If we choose the third row, then we have . So, we get
Let us review an important aspect of this example. In the end, the calculation of the determinant was just multiplying the three entries along the main diagonal together. As it turns out, the reason that the calculation was so straightforward is because the matrix was an upper triangular matrix. Let us recall the definition of this type of matrix.
Definition: Triangular Matrix
If the entries below the main diagonal are zero, the matrix is an upper triangular matrix.
If the entries above the main diagonal are zero, the matrix is a lower triangular matrix.
Upper and lower triangular matrices are shown:
A matrix is triangular if it is either upper or lower triangular (or both).
The reason that finding determinants of triangular matrices is so simple is that the zeros in one half of the matrix remove much of the calculation. To see this, let us consider the calculation of the determinant for a general upper triangular matrix, using the cofactor expansion along the third row:
In other words, the end result is just the product of the three entries on the main diagonal. Similarly, for lower triangular matrices, the cofactor expansion along the first row gives us
This gives us the following property.
Property: Determinants of Triangular Matrices
The determinant of a triangular matrix is the product of the entries on the main diagonal:
As a side note, this property also encompasses the subclass of triangular matrices: diagonal matrices. Recall that a diagonal matrix is one where only the entries on the main diagonal are nonzero. Because diagonal matrices are both upper and lower triangular matrices at the same time, they naturally exhibit the same property:
This also applies to identity matrices (where the product of the diagonal entries is always 1) and zero matrices (where the product is always 0), since these are both special cases of diagonal, and hence triangular, matrices.
Let us see how we can use this property to simplify our solutions in the next example.
Example 3: Comparing the Values of the Determinants of Two Triangular Matrices
True or False: If then .
One way to answer this question would be to calculate each determinant using the cofactor expansion along the rows or columns. However, we can more efficiently answer this question if we take note of the fact that is an upper triangular matrix and is a lower triangular matrix. We can see this because in , the entries below the main diagonal are zero, and in , the entries above the main diagonal are zero:
Thus, we can use the property that the determinant of a triangular matrix is equal to the product of the entries on the main diagonal. As we can see, and have the same diagonal entries. Therefore,
That is, , so the answer is true.
Let us further explore examples where we need to find the determinants of triangular matrices. In some cases, the easy part will be identifying that a matrix is triangular and applying the property for determinants and the hard part will involve further calculations to reach the required answer.
Example 4: Solving Equations by Finding the Determinant of a Diagonal Matrix
Consider the equation
Determine the value of .
The first thing to notice when we see this matrix is that it is diagonal, which means that all of the entries not on the main diagonal are zero:
Diagonal matrices are a special kind of triangular matrix, and we can recall that the determinant of such a matrix is found by taking the product of the entries on the main diagonal. Therefore, the determinant is
Now, we want to find , using the fact that this determinant is equal to 2. That is,
We can find from here by using the properties of indices, specifically that . By rearranging and squaring both sides, we have
For our final example, let us find the determinant of a matrix given in terms of three variables that we will have to find by finding the determinants of smaller matrices.
Example 5: Finding the Value of a Determinant Involving Unknowns Using Properties
If , , and , find
Since we have been given several equations with determinants in them and three unknown variables, the most obvious thing to do would be to find these determinants and see whether this gives us any information about the variables, so let us do this.
First of all, for the matrices, we can find their determinants using the formula
For the first equation, we have
For the second equation, we have
Finally, the third equation gives us
We could use these equations by themselves to find , , and , but this may be more work than necessary. Let us first find the determinant so we can see what information is required of us. We can simplify the calculation of this matrix by noticing that it is an upper triangular matrix, since the entries below the main diagonal are zero:
Therefore, the determinant will just be the product of the entries on the main diagonal, giving us
To find , note that we can take the product of , , and (since we have already computed these values) and then take the square root. That is to say that we have
Then, by taking the square root, we get
We should be aware that both positive and negative 36 are possibilities here. These different values occur due to the different possible values of , , and , so the value of the determinant will depend on the values of the variables.
Therefore, the determinant is either or 36.
Let us consider the main things we have learned in this explainer.
- We can simplify the calculation of determinants in certain cases if some of the entries are
zero. In particular, we can do so in the following:
- If a matrix has a zero row or column, then its determinant is zero:
- If a matrix is upper triangular, lower triangular, or diagonal, then its determinant is the product of the entries on the main diagonal: