Lesson Explainer: Properties of Determinants

In this explainer, we will learn how to identify the properties of determinants and use them to solve problems.

The determinant of a square matrix is a useful number that can help us determine information about that matrix and can help us solve equations involving matrices. Although we can find the determinant of any square matrix, in this explainer we will focus solely on and matrices.

Let’s start by recalling how to calculate the determinants of these two different sizes of matrices.

Definition: Determinant of a Two-by-Two Matrix

The determinant of a matrix (written as or ) is the difference in the product of the elements on its diagonals.

For the matrix , its determinant is given by

We can find the determinant of matrices in a similar manner.

Definition: Determinant of a Three-by-Three Matrix

If then we can calculate the determinant of by expanding over row , or by expanding over column , where is the matrix minor of found by removing row and column from matrix .

This can be extended to any square matrix. The formula for evaluating the determinant can involve a lot of calculations; this means it can be easy to make mistakes. Using the definition of a determinant, we can state and prove some useful properties that make it easier to find the value of a determinant.

We will start by listing the properties of the determinant before going into detail about each.

Properties: Determinants of Matrices

If is any square matrix of order and , then
If is any square matrix, then
If and are square matrices of the same order, then
The determinant of any square triangular matrix is the product of all of the elements on its main diagonal.

It is important to realize that every property we will discuss works for any square matrix, regardless of its size. However, for the purposes of this explainer, we will only work with and matrices.

It is possible to prove all four of the properties from the definition of a determinant. We will prove these properties for lower order matrices.

We can start with the first property. Let and be a scalar value. Then, the determinant of is given by

We can also prove this for a matrix and scalar , calculating the determinant by expanding over the first row as follows:

We can then factor out the value of and use the definition of the determinant of :

The proof for higher-order matrices is very similar.

For the second property, if , then the transpose of is given by . We can evaluate the determinant of the transpose as follows:

For the third property, we will let and .

We can then find as follows:

Hence, we can find the determinant of as follows:

We can then factor and simplify as follows:

Finally, we will show the fourth property for a lower triangular matrix. However, this property holds for upper triangular matrices and square triangular matrices of any other order.

If is a lower triangular matrix such that then we can evaluate the determinant of by expanding over the first row as follows:

Let’s see some examples of how we can use these properties to evaluate the determinants of matrices and solve problems involving the determinant of a matrix.

Example 1: Using the Properties of Determinants to Evaluate an Expression

If is a square matrix of order and , then .

Answer

We want to find the value of . We can do this by first finding the value of . Recall that for any square matrix of order and ,

Since is a matrix of order , we can use this to find as follows:

Next, we want to use the properties of the determinant to find an expression for in terms of . is still a matrix of order , so

Finally, we recall that taking the transpose of a square matrix does not alter its determinant.

Hence,

This is option C, .

Example 2: Finding the Determinant of a Matrix from the Determinant of a Product of Matrices

If and , find .

Answer

We recall that If and are square matrices of the same order, then

To apply this property, we notice that exists, so must be a square matrix. Similarly, exists, so is also a square matrix. Let be an matrix and be an matrix. Then, by the properties of matrix multiplication, will have the same number of rows as , , and the same number of columns as , . Since this matrix is square, we must have . Finally, since we can multiply matrix by matrix on the right, we must have , so is also an matrix.

Therefore, which we can rearrange to obtain

Example 3: Evaluating an Expression Using the Properties of Determinants

If , , and the size of and that of is , find the value of using the properties of determinants.

Answer

We recall that if is any square matrix of order and , then . Therefore, since and are matrices, we let , so that

Similarly,

Next, we recall that if and are square matrices of the same order, then

This gives us

Substituting these values into the expression gives us

In our next example, we will evaluate the determinant of a triangular matrix and explore another property of the determinant of matrices.

Example 4: Evaluating the Determinant of a Matrix by Using the Properties of the Determinant

Find the value of

Answer

In this question, we are asked to evaluate the determinant of a given three-by-three matrix.

We can see that every entry in the matrix below the main diagonal is equal to zero:

Therefore, this is an upper triangular matrix and we recall that we can evaluate the determinant of this type of square matrix by taking the product of its main diagonal. So, we have

Hence, the determinant of this matrix is 0.

In the previous example, we found the determinant of a square triangular matrix by finding the product of its main diagonal. However, there is another method we could have used, which is using the properties of determinants. Recall that we can evaluate the determinant of a square matrix by expanding over any row or column. If we expand over the third row, we have

Since there is an entire row of the entry 0, we see the coefficient of each minor is 0 and so the determinant of the matrix is 0. This will be true for any matrix with a row or column of zeros. This gives us the following property: if every entry in a single row or column of a square matrix is zero, then the determinant of matrix is zero.

In our next example, we will use our knowledge of the properties of determinants to find the value of a variable.

Example 5: Evaluating an Expression Using the Determinant of a Diagonal Matrix

Consider the equation

Determine the value of .

Answer

In this question, we are given the determinant of a matrix involving a variable and asked to find the value of . We can do this by finding an expression for the determinant of this matrix. We could also do this by using the definition of a determinant and expanding over a row or column. However, we also notice that every entry not on the principal diagonal of this matrix is equal to 0. In other words, this matrix is diagonal.

We also know the determinant of any square triangular matrix is the product of all the terms on its principal diagonal, where we use the fact that a diagonal matrix is also an upper and lower triangular matrix.

Therefore,

This determinant is equal to 2. Hence,

Squaring both sides of the equation gives us

In our final example, we will use properties of the determinant to evaluate the determinant of a triangular matrix.

Example 6: Evaluating the Determinant of a Triangular Matrix to Determine Variable Values

If , , and , find

Answer

In this question, we are asked to evaluate the determinant of a given matrix. We could do this by using the definition of the determinant. However, we can also notice that every entry below the leading diagonal is equal to zero. In other words, this is an upper triangular matrix.

We recall that the determinant of any square triangular matrix is equal to the product of the entries in its leading diagonal.

Hence,

To determine the value of this expression, we will evaluate the three determinants given in the question. Recall that the determinant of a matrix is the difference in the product of its diagonals. This gives us then and finally

Rearranging these three equations we have

We can notice that the product of these three equations includes the expression , which is the determinant of the matrix:

Hence, the determinant of the triangular matrix is either 36 or .

We will finish by recapping some of the key points of this explainer.

Key Points

The properties of the determinant can simplify the process of evaluating determinants.
For any square matrix of order and ,
For any square matrix ,
For any square matrices and of the same order,
If every entry in a single row or column of a square matrix is zero, then the determinant of matrix is zero.
The determinant of any square triangular matrix is the product of all of the terms on its principal diagonal.