Linear System Analysis: Determinant, Solutions, And Singularity

Hey guys! Let's dive into the fascinating world of linear systems today. We're going to break down a problem step-by-step, making sure we understand each concept thoroughly. Our focus is on analyzing a given linear system, calculating determinants, finding solutions, and determining if the system is singular. So, grab your thinking caps, and let's get started!

Understanding the Linear System and Statements

So, we've got this linear system: 2x - 5y = E. Now, without knowing the value of 'E', we can still analyze several aspects of this system. We have four statements to evaluate, and our mission is to figure out which ones are false. These statements touch on important characteristics of linear systems: the determinant of the coefficient matrix, the values of x and y, and whether the system is singular. To kick things off, let's define these concepts clearly. The determinant of a matrix is a special number that can be computed from the elements of a square matrix. It provides valuable information about the matrix, including whether the system of equations has a unique solution. A singular system is one where the determinant of the coefficient matrix is zero, which implies that the system either has no solution or infinitely many solutions. Understanding these definitions is crucial as we dig deeper into the problem. Remember, the goal here isn't just to find the answers, but to understand the 'why' behind them. This foundational knowledge will help you tackle similar problems with confidence. We'll be using these concepts to debunk or confirm each statement, so make sure you've got a good grasp of them. Let's move on to analyzing the statements one by one and see where they lead us. Stay curious, and let’s crack this together!

Analyzing Statement I: The Determinant of the Coefficient Matrix

The first statement claims that the determinant of the coefficient matrix is 1/12. To verify this, we first need to identify the coefficient matrix. In our linear system 2x - 5y = E, the coefficient matrix is simply a 1x2 matrix: [2 -5]. Now, here’s a crucial point: determinants are only defined for square matrices (matrices with the same number of rows and columns). Since our coefficient matrix is 1x2 (one row and two columns), it's not a square matrix, and therefore, we cannot calculate its determinant in the traditional sense. This is a fundamental concept in linear algebra, and it's essential to recognize when a calculation is even possible. So, right off the bat, statement I is false because it attempts to calculate the determinant of a non-square matrix. This highlights the importance of understanding the basic rules and definitions before diving into calculations. You might be tempted to apply a formula without checking if it's applicable, but that's a surefire way to get the wrong answer. Now, let’s think about why determinants are so important for square matrices. They tell us a lot about the invertibility of the matrix and the uniqueness of solutions in a system of equations. But for non-square matrices, this concept doesn't directly translate. The key takeaway here is: always check the dimensions of your matrix before trying to calculate its determinant. We’ve already identified one false statement, but let’s keep going. The next statements involve the values of x and y, which we'll investigate next. Ready to move on? Let's do it!

Evaluating Statements II and III: The Values of x and y

Statements II and III give us specific values for x and y: x = 5 and y = 20. To check if these values are correct, we need to plug them back into our original equation, 2x - 5y = E. So, let's substitute x = 5 and y = 20 into the equation:

2*(5) - 5*(20) = E 10 - 100 = E -90 = E

This tells us that if x = 5 and y = 20 are solutions, then E must be -90. However, the original problem doesn't specify the value of E. This is a critical piece of information! Without knowing E, we cannot definitively say whether x = 5 and y = 20 are solutions. They could be a solution if E is -90, but they are not necessarily a solution for any value of E. This is a subtle but important point. Linear equations have infinitely many solutions unless we have additional constraints or information. In our case, we only have one equation with two unknowns, which generally means there's a whole range of (x, y) pairs that could satisfy the equation. Think of it like a line on a graph – there are countless points on that line. So, statements II and III are not necessarily true without knowing the value of E. They could be true for a specific value of E, but they are not universally true. This highlights the importance of having enough information to solve a system of equations uniquely. If we had another independent equation, we could potentially solve for x and y definitively. But with just one equation, we're left with a family of solutions. Now, let's move on to the final statement, which deals with the system being singular. This ties back to our earlier discussion about determinants, so let's see how it fits into the puzzle.

Assessing Statement IV: The System is Singular

Statement IV claims that the system is singular. Remember, a system is singular if the determinant of its coefficient matrix is zero. But as we discussed earlier in our analysis of Statement I, our coefficient "matrix" [2 -5] is not a square matrix. Therefore, we can't calculate its determinant using the standard definition. So, how do we determine if this system is singular? Well, the concept of singularity is more relevant for systems with the same number of equations and variables (square systems). In such systems, a zero determinant indicates that the equations are linearly dependent, meaning one equation can be derived from the others, leading to either no solution or infinitely many solutions. In our case, we have one equation with two variables. This system will always have infinitely many solutions as we saw in the previous section when discussing statements II and III. For any value we pick for 'x', we can find a corresponding value for 'y' that satisfies the equation. This situation is different from a singular square system. While it's true that our system doesn't have a unique solution, the term "singular" is not the most accurate way to describe it in this context. Singularity typically refers to square systems with a determinant of zero. Therefore, we can consider statement IV to be false in the way it's typically understood in the context of linear algebra. It's crucial to use the correct terminology and understand the nuances of these concepts. Applying the term "singular" here might be misleading. So, to recap, we've analyzed all four statements. We found that Statement I is definitively false because it attempts to calculate the determinant of a non-square matrix. Statements II and III are not necessarily true without knowing the value of E. And Statement IV, while hinting at the infinite solutions, is also considered false in the traditional sense of singularity. Let's wrap up our findings and identify the correct answer.

Identifying the False Statements and Final Answer

Okay, let's put all the pieces together. We've meticulously analyzed each statement and determined the following:

• Statement I: The determinant of the coefficient matrix is 1/12. - FALSE (Cannot calculate determinant of a non-square matrix)
• Statement II: The value of x is 5. - NOT NECESSARILY TRUE (Depends on the value of E)
• Statement III: The value of y is 20. - NOT NECESSARILY TRUE (Depends on the value of E)
• Statement IV: The system is singular. - FALSE (Singularity typically applies to square systems)

Now, let's look at the answer choices and see which one matches our findings. The question asks which statements are false. Based on our analysis, statements I and IV are definitively false, and statements II and III are not necessarily true without additional information. However, the answer choices typically focus on the statements that are undeniably false based on the given information.

Looking back at the options (which were not provided in the original prompt, but we can infer common answer structures), the correct answer would be the one that lists statements I and IV as false. This is because we can definitively prove these statements to be incorrect using the principles of linear algebra.

So, in conclusion, by carefully examining each statement and applying the correct definitions and concepts, we were able to identify the false statements and understand why they are incorrect. This highlights the importance of a solid foundation in linear algebra and the ability to analyze problems methodically. Great job, guys! We tackled this one like pros.