Solving Matrices: Find 2P + R Where A = C - B

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Hey guys! Today, we're diving into the world of matrices and tackling a fun problem. We've got three matrices, A, B, and C, and we need to figure out the value of 2P + r based on the equation A = C - B. Sounds like a puzzle, right? Let's break it down step by step.

Understanding the Matrix Problem

Before we jump into the solution, let's make sure we're all on the same page about what matrices are and how matrix subtraction works. Think of a matrix as a grid of numbers arranged in rows and columns. In our case, we have 2x2 matrices, meaning they have two rows and two columns. This foundation is crucial, guys, because misunderstanding it can lead to confusion later on. When we subtract matrices, we subtract corresponding elements. That is, the element in the first row and first column of matrix B is subtracted from the element in the first row and first column of matrix C, and so on. Knowing this basic principle, we can ensure that the matrix operations performed are accurate, which will ultimately lead to the correct solution. For example, if we have:

A =  \begin{bmatrix} a & b \\ c & d \end{bmatrix}
B =  \begin{bmatrix} e & f \\ g & h \end{bmatrix}

Then, A - B would be:

A - B = \begin{bmatrix} a-e & b-f \\ c-g & d-h \end{bmatrix}

This simple rule is the backbone of our entire problem, and understanding it thoroughly will make solving for 2P + r much easier. Remember, attention to detail is key in matrix operations. So, as we move forward, keep this foundational concept in mind. Guys, it’s like building with LEGOs; if the base isn't solid, the whole structure might wobble!

Defining Matrices A, B, and C

Okay, let's define our matrices. We're given:

A = \begin{bmatrix} 3 & 1 \\ -1 & p \end{bmatrix}
B = \begin{bmatrix} 7 & 2 \\ 4 & 3 \end{bmatrix}
C = \begin{bmatrix} p+r & 3 \\ 7 & 3 \end{bmatrix}

Notice that matrix A has 'p' as an element, and matrix C has both 'p' and 'r' in it. Our mission, should we choose to accept it, is to find the values of 'p' and 'r' so we can calculate 2P + r. This is where things get interesting! We're not just dealing with plain numbers; we've got variables mixed in, which means we'll need to use some algebraic skills to solve this. Don't worry, it's not as scary as it sounds. The key is to take it one step at a time and focus on what we know for sure. Each element in these matrices plays a crucial role, and understanding their relationships is vital. So, let’s keep these matrices in mind as we move forward and start piecing together the puzzle. Think of it like a detective game where each matrix element is a clue!

Setting up the Equation: A = C - B

Now, let's use the given equation: A = C - B. This equation is the heart of our problem. It tells us that matrix A is the result of subtracting matrix B from matrix C. Guys, this is our roadmap! This equation sets up a direct relationship between the elements of the three matrices, allowing us to create individual equations that we can solve. So, to make this more visual, let’s write out the subtraction explicitly:

\begin{bmatrix} 3 & 1 \\ -1 & p \end{bmatrix} = \begin{bmatrix} p+r & 3 \\ 7 & 3 \end{bmatrix} - \begin{bmatrix} 7 & 2 \\ 4 & 3 \end{bmatrix}

This setup allows us to see exactly which elements correspond to each other. For instance, the top-left element of A (which is 3) corresponds to the result of subtracting the top-left element of B (which is 7) from the top-left element of C (which is p+r). Understanding these correspondences is key to setting up the equations correctly. Each element comparison gives us a piece of the puzzle, and by solving these mini-equations, we'll eventually uncover the values of 'p' and 'r'. Remember, the order of subtraction matters here! C - B is not the same as B - C. Keeping this in mind will prevent common mistakes and help us nail the solution. Guys, we're building a bridge here, and each equation is a solid support!

Solving for p and r

Okay, time to roll up our sleeves and solve for 'p' and 'r'! We're going to use the equation A = C - B and the matrix subtraction rules we just discussed. This is where the real math magic happens! By equating corresponding elements, we can create a system of equations that will help us find the values we need. Remember, each element in a matrix has a specific position, and we need to compare elements in the same positions across the matrices. This meticulous comparison is crucial for accuracy. So, let’s dive in and start extracting those equations!

Creating Equations from Matrix Elements

By subtracting matrix B from C, we get:

\begin{bmatrix} 3 & 1 \\ -1 & p \end{bmatrix} = \begin{bmatrix} (p+r)-7 & 3-2 \\ 7-4 & 3-3 \end{bmatrix}

Which simplifies to:

\begin{bmatrix} 3 & 1 \\ -1 & p \end{bmatrix} = \begin{bmatrix} p+r-7 & 1 \\ 3 & 0 \end{bmatrix}

Now we can equate the corresponding elements to form equations:

  1. 3 = p + r - 7
  2. 1 = 1 (This doesn't help us solve for p or r, but it's good to confirm that it's consistent)
  3. -1 = 3 (This seems contradictory! This indicates there might be a mistake in the original problem statement or the matrices provided.)
  4. p = 0

Guys, notice something crucial here! Equation 3, -1 = 3, is a clear contradiction. This means that there is likely an error in the original problem statement or in the given matrices. In mathematical problems, inconsistencies like this often indicate a mistake that needs to be addressed. It's super important to recognize these contradictions because they prevent us from finding a valid solution. So, before we proceed further, we need to acknowledge this issue. However, for the sake of demonstrating the solution process, let's proceed assuming that this is a typo and see how we would solve for p and r if everything were consistent. We’ll use the equations we have, keeping in mind that the final answer might not be accurate due to this initial error. This is a valuable lesson in problem-solving: always check for consistency! We're moving forward with caution, just like good mathematicians do!

Solving for p and r (Assuming Consistency)

Okay, let's pretend for a moment that there wasn't a contradiction and see how we'd solve for 'p' and 'r'. From our equations, we have:

  1. 3 = p + r - 7
  2. p = 0

We already have p = 0, which is awesome! Now we can substitute this value into the first equation:

3 = 0 + r - 7

Simplify to solve for 'r':

3 = r - 7
r = 3 + 7
r = 10

So, if the problem were consistent, we would have found p = 0 and r = 10. Guys, isn't it satisfying when the pieces start falling into place? Substitution is a powerful tool in algebra, and it's really helpful in solving systems of equations like this. By plugging in the value of 'p' into the other equation, we were able to isolate 'r' and find its value. This step-by-step approach is key to tackling more complex problems as well. Now, remember, we're working under the assumption that the original problem had a typo. But let’s continue with our practice and see what the final calculation would be!

Calculating 2P + r

Alright, we've (potentially) found our values for 'p' and 'r'. Now, the final step is to calculate 2P + r. This is the grand finale, guys! We're taking the values we've worked so hard to find and plugging them into a simple expression. It's like the last piece of a jigsaw puzzle clicking into place. So, let’s put our numbers in and see what we get!

Substituting Values of p and r

We found (assuming consistency) that p = 0 and r = 10. So, let’s substitute these values into the expression 2P + r:

2P + r = 2(0) + 10

This simplifies to:

2P + r = 0 + 10
2P + r = 10

So, based on our (potentially flawed) values, 2P + r = 10. Guys, we did it! We followed the steps, used our matrix skills, and arrived at a final answer. It feels pretty good, right? However, we need to remember that this answer is contingent on the assumption that there was a typo in the original problem. The contradiction we found earlier is still lingering, reminding us to be cautious about our result. This is a crucial point in problem-solving: always acknowledge the limitations of your solution. Even if we’ve done everything correctly from a procedural standpoint, if the initial conditions are flawed, the answer might not be accurate. So, let’s keep this in mind as we wrap up our discussion.

Conclusion: The Importance of Consistency

So, guys, we tackled a matrix problem, navigated through the steps of matrix subtraction, solved for variables, and calculated 2P + r. We’ve covered a lot of ground today! But the biggest takeaway from this exercise isn't just the mechanics of solving matrices. It's the critical importance of checking for consistency in problem statements. The contradiction we found, where -1 = 3, highlights that a problem must be logically sound for a solution to be valid. This is a fundamental principle in mathematics and problem-solving in general. If the initial conditions are contradictory, any solution we arrive at will be questionable. Therefore, it’s always wise to double-check the given information and look for any inconsistencies before diving into the calculations. In real-world scenarios, this could mean reviewing data sets for errors or clarifying assumptions in a business case. By identifying potential issues early on, we can save time and effort and ensure the reliability of our results. Guys, remember, a good problem-solver is not just someone who can perform calculations; it's someone who can think critically and identify potential flaws in the problem itself. That’s the skill that truly sets you apart!

If the matrices were consistent, our final answer would be 2P + r = 10. But always remember to double-check the problem for errors! Keep practicing, guys, and you'll become matrix masters in no time!