Calculating The Determinant Of A System Of Equations

Hey guys! Let's dive into a common problem in linear algebra: finding the determinant of the main matrix of a system of equations. This is super important because it tells us a lot about the system, like whether it has a unique solution, no solution, or infinitely many solutions. We'll break down the example you gave step-by-step, so you can totally nail this concept. Ready to get started? Let's go!

Understanding the Problem: Determinants and Systems of Equations

So, the big question is, what exactly is the determinant, and why should we care? Well, the determinant is a special number calculated from a square matrix. Think of a matrix as a grid of numbers, and the determinant is a single value that summarizes some key properties of that matrix. For systems of linear equations, the determinant of the coefficient matrix (which is the main matrix we're talking about) is especially useful. If the determinant isn't zero, it tells us that the system has a unique solution. If it is zero, then we know a few things can happen: either there are infinitely many solutions, or there are no solutions at all (the system is inconsistent).

In our example, we're given the following system of equations:

• 6x₁ - 2x₂ = 5
• -3x₁ - x₂ = 5

Our task is to find the determinant of the main matrix. Remember that the main matrix is made up of the coefficients of the variables x₁ and x₂. This matrix helps us understand the structure of the equations and determine whether a solution exists. Keep in mind that understanding the determinant is like having a superpower. It allows you to quickly assess the nature of the solutions to linear equations. Now, let's learn how to calculate it using a few simple steps. The determinant helps us understand the nature of solutions to equations, so let's get into it.

Step-by-Step Calculation of the Determinant

Alright, let's roll up our sleeves and calculate the determinant! First, we need to create the coefficient matrix, which we'll call A. The coefficient matrix is constructed by taking the coefficients of x₁ and x₂ from each equation and arranging them in a matrix. From the equations above, the coefficient matrix A will be:

| 6  -2 |
| -3 -1 |

Now, to find the determinant of a 2x2 matrix, we use a straightforward formula. The determinant of a matrix

| a b |
| c d |

is calculated as (a * d) - (b * c). So, for our matrix A, this means:

det(A) = (6 * -1) - (-2 * -3)

Let's break that down, because math can seem scary if we don't take it slow, right?

• Multiply the top-left element (6) by the bottom-right element (-1): 6 * -1 = -6.
• Multiply the top-right element (-2) by the bottom-left element (-3): -2 * -3 = 6.
• Subtract the second result from the first: -6 - 6 = -12.

So, the determinant of the matrix A is -12. This means our system of equations has a unique solution. Easy peasy, right?

Keep in mind that this process is for 2x2 matrices, and the method changes slightly for larger matrices (3x3, 4x4, etc.), but the core idea remains the same: calculate a single value that tells us about the properties of the system.

Interpreting the Result: What Does the Determinant Tell Us?

So, we've found that the determinant of our main matrix is -12. But what does this actually mean in the context of the system of equations? A non-zero determinant (like our -12) tells us several important things. First and foremost, it means that the system of equations has a unique solution. This is great news! It means that there is only one set of values for x₁ and x₂ that satisfies both equations simultaneously. If the determinant were zero, we'd have a different scenario, either no solutions or infinitely many solutions. This would mean that the equations are either inconsistent (no solution) or dependent (infinitely many solutions).

Essentially, the value of the determinant helps us classify the behavior of the system. A determinant of 0 indicates that the system is either dependent or inconsistent. A non-zero determinant, like -12, means that the system is consistent and independent, and the equations intersect at a single point. So, when you're working through these problems, always pay close attention to the value of the determinant to understand the nature of your solution. It's like a secret code that unlocks the behavior of linear equations! Also, the absolute value of the determinant can be related to the area or volume scaling factor of the linear transformation represented by the matrix. So, even though our focus here is on solving equations, the determinant has far-reaching implications in various areas of mathematics and physics.

Conclusion: Summary and Final Answer

Alright, let's wrap things up! We started with a system of two linear equations, created the coefficient matrix, and calculated its determinant. We found that the determinant is -12. This means the system has a unique solution. Looking back at our multiple-choice options, we can see that the correct answer is:

3 -12

So, the determinant of the main matrix for the given system of equations is indeed -12. Congrats, you've successfully navigated through this problem! Now you have the tools to handle similar problems with confidence. Keep practicing and applying these steps, and you'll get better and better at solving linear algebra problems. Remember, the determinant is a powerful concept, and now you have it in your problem-solving toolkit. Now, you should easily be able to calculate the determinant of a 2x2 matrix and understand what the result tells you about the system of equations. Keep up the amazing work!

Key Takeaways:

• The determinant is a number calculated from a square matrix.
• For a 2x2 matrix

  | a b |
  | c d |
  

  the determinant is (a * d) - (b * c).
• A non-zero determinant indicates a unique solution for the system of equations.
• A determinant of 0 indicates either no solutions or infinitely many solutions.

Happy calculating! You got this!