Lesson Explainer: Properties of Determinants Mathematics

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 2Γ—2 and 3Γ—3 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 2Γ—2 matrix (written as |𝐴| or det(𝐴)) is the difference in the product of the elements on its diagonals.

For the matrix ο€Όπ‘Žπ‘π‘π‘‘οˆ, its determinant is given by |||π‘Žπ‘π‘π‘‘|||=π‘Žπ‘‘βˆ’π‘π‘.

We can find the determinant of 3Γ—3 matrices in a similar manner.

Definition: Determinant of a Three-by-Three Matrix

If 𝐴=οπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žο, then we can calculate the determinant of 𝐴 by expanding over row 𝑖, |𝐴|=ο„š(βˆ’1)π‘Ž|𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|,οŠ©ο…οŠ²οŠ§οƒοŠ°ο…οƒο…οƒο…οƒοŠ°οŠ§οƒοŠ§οƒοŠ§οƒοŠ°οŠ¨οƒοŠ¨οƒοŠ¨οƒοŠ°οŠ©οƒοŠ©οƒοŠ© or by expanding over column 𝑗, |𝐴|=ο„š(βˆ’1)π‘Ž|𝐴|=(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|+(βˆ’1)π‘Ž|𝐴|,οŠ©οƒοŠ²οŠ§οƒοŠ°ο…οƒο…οƒο…οŠ§οŠ°ο…οŠ§ο…οŠ§ο…οŠ¨οŠ°ο…οŠ¨ο…οŠ¨ο…οŠ©οŠ°ο…οŠ©ο…οŠ©ο… where 𝐴 is the matrix minor of 𝐴 found by removing row 𝑖 and column 𝑗 from matrix 𝐴.

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

  1. If 𝐴 is any square matrix of order 𝑛×𝑛 and π‘˜βˆˆβ„, then |π‘˜π΄|=π‘˜|𝐴|.
  2. If 𝐴 is any square matrix, then |𝐴|=|𝐴|.
  3. If 𝐴 and 𝐡 are square matrices of the same order, then |𝐴𝐡|=|𝐴||𝐡|.
  4. 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 2Γ—2 and 3Γ—3 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 𝐴=ο€Όπ‘Žπ‘Žπ‘Žπ‘ŽοˆοŠ§οŠ§οŠ§οŠ¨οŠ¨οŠ§οŠ¨οŠ¨ and π‘˜ be a scalar value. Then, the determinant of π‘˜π΄ is given by |π‘˜π΄|=||π‘˜ο€Όπ‘Žπ‘Žπ‘Žπ‘Žοˆ||=|||π‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Ž|||=(π‘˜π‘Ž)(π‘˜π‘Ž)βˆ’(π‘˜π‘Ž)(π‘˜π‘Ž)=π‘˜(π‘Žπ‘Žβˆ’π‘Žπ‘Ž)=π‘˜|𝐴|.

We can also prove this for a 3Γ—3 matrix 𝐴 and scalar π‘˜, calculating the determinant by expanding over the first row as follows: |π‘˜π΄|=||||π‘˜οπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žο||||=|||||π‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Ž|||||=(βˆ’1)π‘˜π‘Ž|||π‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Ž|||+(βˆ’1)π‘˜π‘Ž|||π‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Ž|||+(βˆ’1)π‘˜π‘Ž|||π‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Žπ‘˜π‘Ž|||=(βˆ’1)π‘˜π‘Žο€Ήπ‘˜π‘Žπ‘Žβˆ’π‘˜π‘Žπ‘Žο…+(βˆ’1)π‘˜π‘Žο€Ήπ‘˜π‘Žπ‘Žβˆ’π‘˜π‘Žπ‘Žο…+(βˆ’1)π‘˜π‘Žο€Ήπ‘˜π‘Žπ‘Žβˆ’π‘˜π‘Žπ‘Žο….

We can then factor out the value of π‘˜οŠ© and use the definition of the determinant of 𝐴: |π‘˜π΄|=π‘˜ο’(βˆ’1)π‘Ž(π‘Žπ‘Žβˆ’π‘Žπ‘Ž)+(βˆ’1)π‘Ž(π‘Žπ‘Žβˆ’π‘Žπ‘Ž)+(βˆ’1)π‘Ž(π‘Žπ‘Žβˆ’π‘Žπ‘Ž)=π‘˜|𝐴|.

The proof for higher-order matrices is very similar.

For the second property, if 𝐴=ο€Όπ‘Žπ‘Žπ‘Žπ‘ŽοˆοŠ§οŠ§οŠ§οŠ¨οŠ¨οŠ§οŠ¨οŠ¨, then the transpose of 𝐴 is given by 𝐴=ο€Όπ‘Žπ‘Žπ‘Žπ‘ŽοˆοοŠ§οŠ§οŠ¨οŠ§οŠ§οŠ¨οŠ¨οŠ¨. We can evaluate the determinant of the transpose as follows: |𝐴|=||π‘Žπ‘Žπ‘Žπ‘Ž||=π‘Žπ‘Žβˆ’π‘Žπ‘Ž=π‘Žπ‘Žβˆ’π‘Žπ‘Ž=|𝐴|.

For the third property, we will let 𝐴=ο€Όπ‘Žπ‘Žπ‘Žπ‘ŽοˆοŠ§οŠ§οŠ¨οŠ§οŠ§οŠ¨οŠ¨οŠ¨ and 𝐡=ο€Ύπ‘π‘π‘π‘οŠοŠ§οŠ§οŠ¨οŠ§οŠ§οŠ¨οŠ¨οŠ¨.

We can then find 𝐴𝐡 as follows: 𝐴𝐡=ο€Όπ‘Žπ‘Žπ‘Žπ‘Žοˆο€Ύπ‘π‘π‘π‘οŠ=ο€Ύπ‘Žπ‘+π‘Žπ‘π‘Žπ‘+π‘Žπ‘π‘Žπ‘+π‘Žπ‘π‘Žπ‘+π‘Žπ‘οŠ.

Hence, we can find the determinant of 𝐴𝐡 as follows: |𝐴𝐡|=|||π‘Žπ‘+π‘Žπ‘π‘Žπ‘+π‘Žπ‘π‘Žπ‘+π‘Žπ‘π‘Žπ‘+π‘Žπ‘|||=(π‘Žπ‘+π‘Žπ‘)(π‘Žπ‘+π‘Žπ‘)βˆ’(π‘Žπ‘+π‘Žπ‘)(π‘Žπ‘+π‘Žπ‘)=π‘Žπ‘π‘Žπ‘+π‘Žπ‘π‘Žπ‘+π‘Žπ‘π‘Žπ‘+π‘Žπ‘π‘Žπ‘βˆ’π‘Žπ‘π‘Žπ‘βˆ’π‘Žπ‘π‘Žπ‘βˆ’π‘Žπ‘π‘Žπ‘βˆ’π‘Žπ‘π‘Žπ‘=π‘Žπ‘π‘Žπ‘+π‘Žπ‘π‘Žπ‘βˆ’π‘Žπ‘π‘Žπ‘βˆ’π‘Žπ‘π‘Žπ‘.

We can then factor and simplify as follows: |𝐴𝐡|=π‘Žπ‘π‘Žπ‘+π‘Žπ‘π‘Žπ‘βˆ’π‘Žπ‘π‘Žπ‘βˆ’π‘Žπ‘π‘Žπ‘=π‘Žπ‘Ž(π‘π‘βˆ’π‘π‘)βˆ’π‘Žπ‘Ž(βˆ’π‘π‘+𝑏𝑏)=(π‘Žπ‘Žβˆ’π‘Žπ‘Ž)(π‘π‘βˆ’π‘π‘)=|𝐴||𝐡|.

Finally, we will show the fourth property for a 3Γ—3 lower triangular matrix. However, this property holds for upper triangular matrices and square triangular matrices of any other order.

If 𝐴 is a 3Γ—3 lower triangular matrix such that 𝐴=ο‚π‘Ž00π‘Žπ‘Ž0π‘Žπ‘Žπ‘ŽοŽ, then we can evaluate the determinant of 𝐴 by expanding over the first row as follows: |𝐴|=π‘Ž|𝐴|βˆ’π‘Ž|𝐴|+π‘Ž|𝐴|=π‘Ž|||π‘Ž0π‘Žπ‘Ž|||βˆ’0|||π‘Ž0π‘Žπ‘Ž|||+0|||π‘Žπ‘Žπ‘Žπ‘Ž|||=π‘Ž|||π‘Ž0π‘Žπ‘Ž|||=π‘Ž(π‘Žπ‘Žβˆ’0Γ—π‘Ž)=π‘Žπ‘Žπ‘Ž.

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 𝐴 is a square matrix of order 2Γ—2 and |2𝐴|=12, then |3𝐴|=.

  1. 18
  2. 24
  3. 27
  4. 36

Answer

We want to find the value of |3𝐴|. We can do this by first finding the value of |𝐴|. Recall that for any square matrix 𝐴 of order 𝑛×𝑛 and π‘˜βˆˆβ„, |π‘˜π΄|=π‘˜|𝐴|.

Since 𝐴 is a matrix of order 2Γ—2, we can use this to find |𝐴| as follows: |2𝐴|=122|𝐴|=12|𝐴|=3.

Next, we want to use the properties of the determinant to find an expression for |3𝐴| in terms of |𝐴|. 𝐴 is still a matrix of order 2Γ—2, so |3𝐴|=3|𝐴|=9|𝐴|.

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

Hence, 9|𝐴|=9|𝐴|=9(3)=27.

This is option C, |3𝐴|=27.

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

If det(𝐴𝐡)=18 and det(𝐴)=2, find det(𝐡).

Answer

We recall that If 𝐴 and 𝐡 are square matrices of the same order, then detdetdet(𝐴𝐡)=(𝐴)Γ—(𝐡).

To apply this property, we notice that det(𝐴) exists, so 𝐴 must be a square matrix. Similarly, det(𝐴𝐡) exists, so 𝐴𝐡 is also a square matrix. Let 𝐴 be an 𝑛×𝑛 matrix and 𝐡 be an π‘šΓ—π‘™ matrix. Then, by the properties of matrix multiplication, 𝐴𝐡 will have the same number of rows as 𝐴, 𝑛, and the same number of columns as 𝐡, 𝑙. Since this matrix is square, we must have 𝑛=𝑙. Finally, since we can multiply matrix 𝐴 by matrix 𝐡 on the right, we must have 𝑛=π‘š, so 𝐡 is also an 𝑛×𝑛 matrix.

Therefore, 18=(𝐴𝐡)=(𝐴)Γ—(𝐡)=2Γ—(𝐡),detdetdetdet which we can rearrange to obtain det(𝐡)=9.

Example 3: Evaluating an Expression Using the Properties of Determinants

If det(𝐴)=2, det(𝐡)=3, and the size of 𝐴 and that of 𝐡 is 2Γ—2, find the value of detdetdet(2𝐴)+(3𝐡)+(𝐴𝐡) using the properties of determinants.

Answer

We recall that if 𝐴 is any square matrix of order 𝑛×𝑛 and π‘˜βˆˆβ„, then detdet(π‘˜π΄)=π‘˜(𝐴). Therefore, since 𝐴 and 𝐡 are 2Γ—2 matrices, we let 𝑛=2, so that detdet(2𝐴)=2(𝐴)=4Γ—2=8.

Similarly, detdet(3𝐡)=3(𝐡)=9Γ—3=27.

Next, we recall that if 𝐴 and 𝐡 are square matrices of the same order, then detdetdet(𝐴𝐡)=(𝐴)(𝐡).

This gives us det(𝐴𝐡)=2Γ—3=6.

Substituting these values into the expression gives us detdetdet(2𝐴)+(3𝐡)+(𝐴𝐡)=8+27+6=41.

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 ||||5βˆ’1βˆ’80260000||||.

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: ||||5βˆ’1βˆ’80260000||||.

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 ||||5βˆ’1βˆ’80260000||||=5Γ—2Γ—0=0.

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 ||||41βˆ’8036000||||=0||1βˆ’836||βˆ’0||4βˆ’806||+0||4103||=0.

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 𝐴 is zero, then the determinant of matrix 𝐴 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 ||||π‘₯βˆ’1000π‘₯+π‘₯+10001||||=2.

Determine the value of π‘₯.

Answer

In this question, we are given the determinant of a 3Γ—3 matrix involving a variable π‘₯ and asked to find the value of π‘₯. 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, ||||π‘₯βˆ’1000π‘₯+π‘₯+10001||||=(π‘₯βˆ’1)ο€Ήπ‘₯+π‘₯+1(1)=π‘₯+π‘₯+π‘₯βˆ’π‘₯βˆ’π‘₯βˆ’1=π‘₯βˆ’1.

This determinant is equal to 2. Hence, π‘₯βˆ’1=2π‘₯=3.

Squaring both sides of the equation gives us ο€Ήπ‘₯=3π‘₯=9.

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 detο€½π‘₯44𝑦=0, det𝑦99𝑧=0, and detο€Όπ‘₯11π‘§οˆ=0, find detπ‘₯120𝑦300𝑧.

Answer

In this question, we are asked to evaluate the determinant of a given 3Γ—3 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, detπ‘₯120𝑦300𝑧=π‘₯𝑦𝑧.

To determine the value of this expression, we will evaluate the three determinants given in the question. Recall that the determinant of a 2Γ—2 matrix is the difference in the product of its diagonals. This gives us 0=ο€½π‘₯44𝑦0=π‘₯π‘¦βˆ’16,det then 0=𝑦99𝑧0=π‘¦π‘§βˆ’81,det and finally 0=ο€Όπ‘₯11π‘§οˆ0=π‘₯π‘§βˆ’1.det

Rearranging these three equations we have π‘₯𝑦=16,𝑦𝑧=81,π‘₯𝑧=1.

We can notice that the product of these three equations includes the expression π‘₯𝑦𝑧, which is the determinant of the 3Γ—3 matrix: (π‘₯𝑦)Γ—(𝑦𝑧)Γ—(π‘₯𝑧)=16Γ—81Γ—1(π‘₯𝑦𝑧)=1296π‘₯𝑦𝑧=Β±36.

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

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 𝐴 of order 𝑛×𝑛 and π‘˜βˆˆβ„, |π‘˜π΄|=π‘˜|𝐴|.
  • For any square matrix 𝐴, |𝐴|=|𝐴|.
  • For any square matrices 𝐴 and 𝐡 of the same order, |𝐴𝐡|=|𝐴||𝐡|.
  • If every entry in a single row or column of a square matrix 𝐴 is zero, then the determinant of matrix 𝐴 is zero.
  • The determinant of any square triangular matrix is the product of all of the terms on its principal diagonal.

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