Lesson Explainer: Two-by-Two Determinants Mathematics • 10th Grade

In this explainer, we will learn how to identify determinants and evaluate 2Γ—2 determinants.

We have seen that there are a lot of different operations we can perform on matrices; we can add and subtract them, and we have learned how to multiply a matrix by a scalar. Now, we are going to introduce one of the most important definitions of a matrix: its determinant.

First, it is important to know that not all matrices have a determinant. In fact, only square matrices have a determinant. Second although it is possible to define the determinant of any square matrix, we are only going to cover the case of 2Γ—2 matrices in this explainer.

Definition: Two-by-Two Determinants

The determinant of a 2Γ—2 matrix 𝐴 (written |𝐴|) is the difference in the product of its diagonals.

For example, |||π‘Žπ‘π‘π‘‘|||=π‘Žπ‘‘βˆ’π‘π‘.

We can also write the determinant as detο€Όπ‘Žπ‘π‘π‘‘οˆ, since it could be easy to mistake the vertical bar notation with the absolute value of a number. In our first few examples, we will demonstrate how to calculate the determinant of different matrices.

Example 1: Evaluating the Determinant of a Given Matrix

Find the determinant of the matrix ο€Ό51βˆ’15.

Answer

First, we can see that the matrix given to us is a square 2Γ—2 matrix.

We know that we can calculate the determinant of a matrix 𝐴=ο€Όπ‘Žπ‘π‘π‘‘οˆ as |||π‘Žπ‘π‘π‘‘|||=π‘Žπ‘‘βˆ’π‘π‘.

So, to calculate the determinant of our matrix, we need to find the difference in the product of its diagonals.

The product of our main diagonal is 5Γ—5=25.

The product of our other diagonal is 1Γ—(βˆ’1)=βˆ’1.

This means we can calculate the determinant as follows: ||51βˆ’15||=5Γ—5βˆ’1Γ—(βˆ’1)=25βˆ’(βˆ’1)=26.

Therefore, the determinant of ο€Ό51βˆ’15 is equal to 26.

Example 2: Evaluating a Determinant

Find the value of ||βˆ’2βˆ’977||.

Answer

In this question, we are asked to find the determinant of a 2Γ—2 matrix. To do this, we need to find the difference in the product of its diagonals.

In other words, if 𝐴=ο€Όπ‘Žπ‘π‘π‘‘οˆ, then |||π‘Žπ‘π‘π‘‘|||=π‘Žπ‘‘βˆ’π‘π‘.

The product of the main diagonal of this matrix is βˆ’2Γ—7=βˆ’14 and the product of the other diagonal of this matrix is βˆ’9Γ—7=βˆ’63.

Then, ||βˆ’2βˆ’977||=(βˆ’2)Γ—7βˆ’(βˆ’9)Γ—7=βˆ’14βˆ’(βˆ’63)=49.

Therefore, we have shown that ||βˆ’2βˆ’977||=49.

Now that we have defined the determinant, we can calculate the determinant of a few matrices we already know: |0|=||0000||=0Γ—0βˆ’0Γ—0=0,|𝐼|=||1001||=1Γ—1βˆ’0Γ—0=1.

There is also one more useful property of the determinant of a matrix that we can show.

If 𝐴=ο€Όπ‘Žπ‘π‘π‘‘οˆ, then the transpose of 𝐴 is given by 𝐴=ο€»π‘Žπ‘π‘π‘‘ο‡οŒ³. We can then use this to calculate the determinant of the transpose of 𝐴: |𝐴|=||π‘Žπ‘π‘π‘‘||=π‘Žπ‘‘βˆ’π‘π‘=π‘Žπ‘‘βˆ’π‘π‘=|𝐴|.

Let us summarize the results we have just shown about determinants of 2Γ—2 matrices.

Definition: Properties of the Determinant of Two-by-Two Matrices

  • The determinant of the 2Γ—2 zero matrix is 0: ||0000||=0.
  • The determinant of the 2Γ—2 identity matrix is 1: ||1001||=1.
  • Transposing a 2Γ—2 matrix does not change the value of its determinant: |||π‘Žπ‘π‘π‘‘|||=||π‘Žπ‘π‘π‘‘||=π‘Žπ‘‘βˆ’π‘π‘.

Sometimes, we will need to use our definition of the determinant to find the values of unknowns or to solve equations. Let us see some examples of how we can do this.

Example 3: Evaluating a Determinant in terms of π‘₯

Find the value of the determinant 𝐴=||π‘₯βˆ’11π‘₯βˆ’1||.

Answer

We are told that 𝐴 is the determinant of a 2Γ—2 matrix. To evaluate this determinant, we need to recall that |||π‘Žπ‘π‘π‘‘|||=π‘Žπ‘‘βˆ’π‘π‘.

Applying this to the matrix given to us in the question, we get ||π‘₯βˆ’11π‘₯βˆ’1||=π‘₯Γ—(βˆ’1)βˆ’(βˆ’11)Γ—π‘₯=βˆ’π‘₯+11π‘₯=10π‘₯.

Therefore, 𝐴=||π‘₯βˆ’11π‘₯βˆ’1||=10π‘₯.

In our next example, we will demonstrate the fact that two matrices that are not equal can have the same determinant.

Example 4: Solving a Simple Equation with Determinants

Fill in the blank: If ||1π‘₯π‘₯3||=||2143||, then π‘₯=.

  1. 1
  2. 2
  3. 1,βˆ’1
  4. √5

Answer

In this question, we are told that the determinants of two different 2Γ—2 matrices are equal and then asked to find the value of the unknown π‘₯. To do this, we will calculate each determinant separately.

Recall that |||π‘Žπ‘π‘π‘‘|||=π‘Žπ‘‘βˆ’π‘π‘.

Let us start with the determinant on the left-hand side of the equation: ||1π‘₯π‘₯3||=1Γ—3βˆ’π‘₯Γ—π‘₯=3βˆ’π‘₯.

Next, we calculate the determinant on the right-hand side of the equation: ||2143||=2Γ—3βˆ’1Γ—4=6βˆ’4=2.

We are told in the question that these two determinants should be equal; this gives us 3βˆ’π‘₯=2.

We can then solve this for π‘₯: π‘₯=1π‘₯=Β±1.

This means that for the determinants of these two matrices to be equal, either π‘₯=1 or π‘₯=βˆ’1.

The answer is option C.

It is also possible that our matrices will contain functions and this will mean we will need to use other identities and techniques from other areas of math to evaluate these determinants. Let us see some examples of this.

Example 5: Applying Trigonometric Identities to Evaluate a Determinant

Evaluate |||10π‘₯βˆ’2π‘₯10π‘₯2π‘₯|||cossinsincos.

Answer

We are tasked with evaluating the determinant of a 2Γ—2 matrix, for which each entry is a trigonometric function.

We calculate the determinant of this matrix by finding the difference in the product of the diagonals: |||10π‘₯βˆ’2π‘₯10π‘₯2π‘₯|||=10π‘₯Γ—2π‘₯βˆ’(βˆ’2π‘₯)Γ—10π‘₯=20π‘₯+20π‘₯=20ο€Ίπ‘₯+π‘₯.cossinsincoscoscossinsincossincossin

Recall that the Pythagorean identity tells us cossinπ‘₯+π‘₯≑1.

We can use this to simplify the expression we found for the determinant: 20ο€Ίπ‘₯+π‘₯=20.cossin

Therefore, |||10π‘₯βˆ’2π‘₯10π‘₯2π‘₯|||=20.cossinsincos

In our next example, we will demonstrate how to extend this idea to find the determinant of a matrix whose elements are reciprocal trigonometric functions.

Example 6: Solving Trigonometric Equations Involving Determinants

Solve the equation |||βˆ’πœƒπœƒπœƒπœƒ|||=βˆ’2cossincsccsc given that 0<πœƒ<90∘∘.

Answer

We need to solve an equation that involves the determinant of a 2Γ—2 matrix.

We calculate the determinant of this matrix by finding the difference in the product of its diagonals: |||βˆ’πœƒπœƒπœƒπœƒ|||=βˆ’πœƒΓ—πœƒβˆ’(πœƒ)Γ—πœƒ.cossincsccsccoscscsincsc

We can simplify this by recalling the following trigonometric identity: cscsinπœƒβ‰‘1πœƒ.

Then, |||βˆ’πœƒπœƒπœƒπœƒ|||=βˆ’πœƒΓ—πœƒβˆ’(πœƒ)Γ—πœƒ=βˆ’πœƒΓ—1πœƒβˆ’(πœƒ)Γ—1πœƒ=βˆ’πœƒπœƒβˆ’1.cossincsccsccoscscsincsccossinsinsincossin

To solve the original equation, we will set this expression equal to βˆ’2 for values of πœƒ in the interval 0<πœƒ<90∘∘: βˆ’πœƒπœƒβˆ’1=βˆ’2πœƒπœƒ=1.cossincossin

We can then take the reciprocal to rewrite the equation as tanπœƒ=1.

Recall, tan(45)=1∘; this gives us one solution to our equation in the interval. We still need to check for other possible solutions. We will do this by sketching the graph 𝑦=(πœƒ)tan and the line 𝑦=1.

The solutions to our equation will be the intersections of these lines. The only intersection in the interval 0<πœƒ<90∘∘ is when πœƒ=45∘, so this is our only solution.

Therefore, we have shown that if |||βˆ’πœƒπœƒπœƒπœƒ|||=βˆ’2cossincsccsc and 0<πœƒ<90∘∘, then πœƒ=45.∘

We will finish by recapping some of the key points explored in this explainer on the determinant of a 2Γ—2 matrix.

Key Points

  • The determinant of a 2Γ—2 matrix is calculated by taking the difference of the product of its diagonals: |||π‘Žπ‘π‘π‘‘|||=π‘Žπ‘‘βˆ’π‘π‘.
  • The determinant of the 2Γ—2 zero matrix is 0.
  • The determinant of the 2Γ—2 identity matrix is 1.
  • Transposing a 2Γ—2 matrix does not change its determinant.

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