Lesson Explainer: Two-by-Two Determinants Mathematics

In this explainer, we will learn how to identify determinants and evaluate 2×2 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 2×2 matrices in this explainer.

Definition: Two-by-Two Determinants

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

For example, |||𝑎𝑏𝑐𝑑|||=𝑎𝑑𝑏𝑐.

We can also write the determinant as det𝑎𝑏𝑐𝑑, 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 5115.

Answer

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

We know that we can calculate the determinant of a matrix 𝐴=𝑎𝑏𝑐𝑑 as |||𝑎𝑏𝑐𝑑|||=𝑎𝑑𝑏𝑐.

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 5×5=25.

The product of our other diagonal is 1×(1)=1.

This means we can calculate the determinant as follows: ||5115||=5×51×(1)=25(1)=26.

Therefore, the determinant of 5115 is equal to 26.

Example 2: Evaluating a Determinant

Find the value of ||2977||.

Answer

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

In other words, if 𝐴=𝑎𝑏𝑐𝑑, then |||𝑎𝑏𝑐𝑑|||=𝑎𝑑𝑏𝑐.

The product of the main diagonal of this matrix is 2×7=14 and the product of the other diagonal of this matrix is 9×7=63.

Then, ||2977||=(2)×7(9)×7=14(63)=49.

Therefore, we have shown that ||2977||=49.

Now that we have defined the determinant, we can calculate the determinant of a few matrices we already know: |0|=||0000||=0×00×0=0,|𝐼|=||1001||=1×10×0=1.

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

If 𝐴=𝑎𝑏𝑐𝑑, then the transpose of 𝐴 is given by 𝐴=𝑎𝑐𝑏𝑑. We can then use this to calculate the determinant of the transpose of 𝐴: |𝐴|=||𝑎𝑐𝑏𝑑||=𝑎𝑑𝑐𝑏=𝑎𝑑𝑏𝑐=|𝐴|.

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

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

  • The determinant of the 2×2 zero matrix is 0: ||0000||=0.
  • The determinant of the 2×2 identity matrix is 1: ||1001||=1.
  • Transposing a 2×2 matrix does not change the value of its determinant: |||𝑎𝑏𝑐𝑑|||=||𝑎𝑐𝑏𝑑||=𝑎𝑑𝑏𝑐.

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 𝑥

Find the value of the determinant 𝐴=||𝑥11𝑥1||.

Answer

We are told that 𝐴 is the determinant of a 2×2 matrix. To evaluate this determinant, we need to recall that |||𝑎𝑏𝑐𝑑|||=𝑎𝑑𝑏𝑐.

Applying this to the matrix given to us in the question, we get ||𝑥11𝑥1||=𝑥×(1)(11)×𝑥=𝑥+11𝑥=10𝑥.

Therefore, 𝐴=||𝑥11𝑥1||=10𝑥.

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

If ||1𝑥𝑥3||=||2143||, then 𝑥=.

  1. 1
  2. 2
  3. 1,1
  4. 5

Answer

In this question, we are told that the determinants of two different 2×2 matrices are equal and then asked to find the value of the unknown 𝑥. To do this, we will calculate each determinant separately.

Recall that |||𝑎𝑏𝑐𝑑|||=𝑎𝑑𝑏𝑐.

Let us start with the determinant on the left-hand side of the equation: ||1𝑥𝑥3||=1×3𝑥×𝑥=3𝑥.

Next, we calculate the determinant on the right-hand side of the equation: ||2143||=2×31×4=64=2.

We are told in the question that these two determinants should be equal; this gives us 3𝑥=2.

We can then solve this for 𝑥: 𝑥=1𝑥=±1.

This means that for the determinants of these two matrices to be equal, either 𝑥=1 or 𝑥=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 |||10𝑥2𝑥10𝑥2𝑥|||cossinsincos.

Answer

We are tasked with evaluating the determinant of a 2×2 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: |||10𝑥2𝑥10𝑥2𝑥|||=10𝑥×2𝑥(2𝑥)×10𝑥=20𝑥+20𝑥=20𝑥+𝑥.cossinsincoscoscossinsincossincossin

Recall that the Pythagorean identity tells us cossin𝑥+𝑥1.

We can use this to simplify the expression we found for the determinant: 20𝑥+𝑥=20.cossin

Therefore, |||10𝑥2𝑥10𝑥2𝑥|||=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 |||𝜃𝜃𝜃𝜃|||=2cossincsccsc given that 0<𝜃<90.

Answer

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

We calculate the determinant of this matrix by finding the difference in the product of its diagonals: |||𝜃𝜃𝜃𝜃|||=𝜃×𝜃(𝜃)×𝜃.cossincsccsccoscscsincsc

We can simplify this by recalling the following trigonometric identity: cscsin𝜃1𝜃.

Then, |||𝜃𝜃𝜃𝜃|||=𝜃×𝜃(𝜃)×𝜃=𝜃×1𝜃(𝜃)×1𝜃=𝜃𝜃1.cossincsccsccoscscsincsccossinsinsincossin

To solve the original equation, we will set this expression equal to 2 for values of 𝜃 in the interval 0<𝜃<90: 𝜃𝜃1=2𝜃𝜃=1.cossincossin

We can then take the reciprocal to rewrite the equation as tan𝜃=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=(𝜃)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<𝜃<90 is when 𝜃=45, so this is our only solution.

Therefore, we have shown that if |||𝜃𝜃𝜃𝜃|||=2cossincsccsc and 0<𝜃<90, then 𝜃=45.

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

Key Points

  • The determinant of a 2×2 matrix is calculated by taking the difference of the product of its diagonals: |||𝑎𝑏𝑐𝑑|||=𝑎𝑑𝑏𝑐.
  • The determinant of the 2×2 zero matrix is 0.
  • The determinant of the 2×2 identity matrix is 1.
  • Transposing a 2×2 matrix does not change its determinant.

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