Solving Complex Equations: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a fun problem involving complex numbers and determinants. We'll walk through how to solve for x and y in the given equation: $\left|\begin{array}{ccc} 1 & 1+2 j & 0 \ 1-j & -1 & -1 \ x & 0 & y j \end{array}\right|=2 j$. Don't worry, if you're new to complex numbers or determinants, we'll break it down step by step. Let's get started!

Understanding the Problem: Complex Numbers and Determinants

Alright, before we jump into the solution, let's make sure we're all on the same page. This problem is all about complex numbers and determinants. So, what are they? Well, a complex number is a number that can be expressed in the form a + bj, where a and b are real numbers, and j is the imaginary unit, defined as the square root of -1 ($\sqrt{-1}$). Basically, it's a number that includes a real part (a) and an imaginary part (b). In our problem, you'll see complex numbers like 1 + 2j and 1 - j. These are the building blocks we'll be working with. On the other hand, the determinant is a value that can be computed from the elements of a square matrix. Think of it as a special number that we get when we do some calculations with the matrix. The determinant provides useful information about the matrix, like whether the matrix has an inverse or not. So, we're given a 3x3 matrix, and we need to figure out the determinant. Then, use the information provided to find the values of x and y. The equation gives us a starting point and tells us what the determinant should equal, so we can do the math correctly. Remember, the goal here is to find the values of x and y that make the determinant of the given matrix equal to 2j. This is like a puzzle. We have all the pieces, we know what the finished picture looks like, and we just need to put them together to find x and y. The first step will be computing the determinant.

Calculating the Determinant: The Key to the Solution

Okay, guys, let's get our hands dirty and actually calculate the determinant of the given matrix. There are several ways to compute a determinant, but we'll use the cofactor expansion method. It's a straightforward approach, especially for a 3x3 matrix. Here's how it works: First, we choose a row or column to expand along. For simplicity, let's expand along the first row. The determinant is calculated as follows: $\left|\begin{array}{ccc} 1 & 1+2 j & 0 \ 1-j & -1 & -1 \ x & 0 & y j \end{array}\right| = 1 \cdot C_{11} - (1+2j) \cdot C_{12} + 0 \cdot C_{13}$. Where Cij are the cofactors, which are computed by taking the determinant of the smaller matrix. Now, let's figure out the cofactors. The cofactor C11 is the determinant of the matrix formed by removing the first row and first column: C11 = (-1)(yj) - (-1)(0) = -yj. The cofactor C12 is the determinant of the matrix formed by removing the first row and second column, multiplied by -1: C12 = -[(1-j)(yj) - (-1)(x)] = -[yj - j^2y + x] = -[yj + y + x]. And lastly, the cofactor C13 is the determinant of the matrix formed by removing the first row and third column: C13 = (1-j)(0) - (-1)(x) = x. Okay, guys, now we have all the necessary values, so we can put it all together: The determinant is: $1 \cdot (-yj) - (1+2j) \cdot (-(yj + y + x)) + 0 \cdot x = -yj + (1+2j)(yj + y + x)$. Now, this looks a little messy, but don't worry, we will simplify it step by step. Let's expand the expression: $-yj + yj + y + x + 2j^2y + 2jy + 2jx$. Since j is the square root of -1, we know that j^2 = -1. Substituting this into the equation, we get: $-yj + yj + y + x - 2y + 2jy + 2jx$. Simplifying it, we obtain: $-y + x + 2jy + 2jx$. We can further rewrite this as $(x-y) + (2x+2y)j$. Awesome, we have found the determinant, but we are not done yet!

Solving for x and y: Putting it all Together

Alright, now we've done the heavy lifting and calculated the determinant. Remember, the original equation tells us that the determinant equals 2j. So, we now have: $(x-y) + (2x+2y)j = 2j$. Now we can equate the real and imaginary parts separately. For the real part, we have: x - y = 0. For the imaginary part, we have: 2x + 2y = 2. This gives us a system of two equations with two variables. From the first equation, x - y = 0, we can easily derive that x = y. Now, we substitute x with y in the second equation: 2y + 2y = 2. Which simplifies to 4y = 2. Dividing both sides by 4, we get y = 1/2. Now that we have found y, we can find x. Since x = y, then x = 1/2. Congratulations! We have found the solution. The values that satisfy the original equation are x = 1/2 and y = 1/2. This is what we were looking for. Now, it's always a good idea to check your answer. To verify this, let's plug these values back into the determinant equation to see if it holds true. We're going to substitute x = 1/2 and y = 1/2 into the determinant: $(1/2 - 1/2) + (2(1/2) + 2(1/2))j = 0 + (1 + 1)j = 2j$. As we can see, the result is indeed 2j, so our solution is correct. Keep in mind, math is all about practice, so don't be discouraged if you don't get it immediately. The more you practice, the better you'll become.

Final Thoughts and Key Takeaways

Guys, we did it! We successfully solved for x and y in the complex determinant equation. We started with understanding the problem, then computed the determinant using the cofactor expansion method. Finally, we solved the system of equations derived from equating the real and imaginary parts. The key takeaways from this problem are understanding complex numbers and determinants, the importance of careful calculation, and how to solve a system of equations. This is also a great example of how different areas of math connect. Determinants, complex numbers, and solving systems of equations are all interconnected. This problem really demonstrates that math is a step-by-step process. By breaking down a complex problem into smaller, manageable steps, we can solve it effectively. Every single step is important, so don't skip any! Remember to practice. Working through similar problems will help you strengthen your skills and confidence. Don't be afraid to make mistakes; they are part of the learning process. Each mistake is a learning opportunity. Keep practicing, keep learning, and keep exploring the fascinating world of mathematics!