Matrix Multiplication: Evaluating PQ Step-by-Step

Hey math enthusiasts! Let's dive into the fascinating world of matrix multiplication. Today, we're going to tackle a specific problem: evaluating the product of two matrices, P and Q. Specifically, we have $P=\left[\begin{array}{cc} 2 & 0 \\ 4 & 5 \\ 1 & -1 \end{array}\right]$ and $Q=\left[\begin{array}{ccc} -1 & 0 & 2 \\ 2 & -2 & 3 \end{array}\right]$. The goal here is to determine whether we can actually perform the matrix multiplication PQ and, if so, calculate the resulting matrix. This is a fundamental concept in linear algebra, and understanding how to perform this operation is crucial for many applications, from computer graphics to solving systems of equations. Let's break down the process step by step, ensuring we grasp every detail. Matrix multiplication might seem a bit tricky at first, but once you understand the rules, it becomes a straightforward process. We'll explore the dimensions of the matrices, explain the rules for matrix multiplication, and then walk through the calculation. By the end, you'll be able to confidently evaluate the product of these matrices. So, buckle up, grab your coffee, and let's get started on this math adventure! We'll start with the basics, making sure everyone is on the same page, and then gradually build up to the solution. The key is to understand the rules and apply them correctly. Let's start with examining the dimensions of the matrices. This is the first and most important step to determine if we can even multiply these two matrices.

Matrix Dimensions and Compatibility

Alright, before we get our hands dirty with calculations, we need to check if matrix multiplication is even possible. The dimensions of the matrices are the key here. The dimensions tell us how many rows and columns each matrix has. Remember, the number of columns in the first matrix (P in our case) must equal the number of rows in the second matrix (Q) for multiplication to be defined. So, let's take a look at the dimensions. Matrix P has 3 rows and 2 columns, so its dimensions are 3x2. Matrix Q has 2 rows and 3 columns, giving us dimensions of 2x3. Now, the crucial part: does the number of columns in P match the number of rows in Q? Yes, it does! Matrix P has 2 columns, and matrix Q has 2 rows. This means that we can multiply P and Q. If those numbers didn't match, we'd have to stop right there and say the multiplication isn't possible. It's like trying to fit a square peg into a round hole; it just doesn't work! But in our case, we're good to go. This check is often the most overlooked part, but it's essential to avoid wasting time on calculations that won't yield a result. Think of it as a gatekeeper; it allows us to proceed only if the conditions are met. So, we've cleared the first hurdle. Now that we know multiplication is possible, let's look at the dimensions of the resulting matrix. The resulting matrix will have the same number of rows as the first matrix (P) and the same number of columns as the second matrix (Q). Therefore, the resulting matrix, which we'll call PQ, will be a 3x3 matrix. With this information, we know what to expect and can double-check our work later to make sure we're on the right track. Knowing the dimensions of the result beforehand is a great way to verify your calculations. Let's move on and actually calculate the values inside the matrix.

Performing the Matrix Multiplication: Step-by-Step

Now, let's roll up our sleeves and calculate the elements of the matrix PQ. Matrix multiplication involves taking the dot product of the rows of the first matrix (P) with the columns of the second matrix (Q). If you're not familiar with the dot product, don't worry; it's quite simple! For each element in the resulting matrix PQ, we'll perform this operation. Remember, the resulting matrix will have 3 rows and 3 columns. Let's start with the element in the first row and first column (PQ₁₁). To calculate this element, we take the dot product of the first row of P (2, 0) and the first column of Q (-1, 2). The dot product is calculated as (2 * -1) + (0 * 2) = -2 + 0 = -2. So, PQ₁₁ = -2. Next, let's calculate the element in the first row and second column (PQ₁₂). We'll take the dot product of the first row of P (2, 0) and the second column of Q (0, -2). This gives us (2 * 0) + (0 * -2) = 0 + 0 = 0. Therefore, PQ₁₂ = 0. And then, PQ₁₃: We take the dot product of the first row of P (2, 0) and the third column of Q (2, 3). This is (2 * 2) + (0 * 3) = 4 + 0 = 4, so PQ₁₃ = 4. Now, let's move on to the second row. For the element in the second row and first column (PQ₂₁), we'll take the dot product of the second row of P (4, 5) and the first column of Q (-1, 2). This gives us (4 * -1) + (5 * 2) = -4 + 10 = 6, thus PQ₂₁ = 6. Let's keep going. For PQ₂₂, take the dot product of the second row of P (4, 5) and the second column of Q (0, -2). So, we have (4 * 0) + (5 * -2) = 0 - 10 = -10. Therefore, PQ₂₂ = -10. And for PQ₂₃, the second row of P (4, 5) and the third column of Q (2, 3), so (4 * 2) + (5 * 3) = 8 + 15 = 23, and PQ₂₃ = 23. Finally, let's find the third row of the matrix. For the element in the third row and first column (PQ₃₁), we multiply the third row of P (1, -1) and the first column of Q (-1, 2). This gives (1 * -1) + (-1 * 2) = -1 - 2 = -3. So, PQ₃₁ = -3. Then, for PQ₃₂, we take the third row of P (1, -1) and the second column of Q (0, -2). This is (1 * 0) + (-1 * -2) = 0 + 2 = 2, so PQ₃₂ = 2. And last but not least, for PQ₃₃, we take the dot product of the third row of P (1, -1) and the third column of Q (2, 3). This gives (1 * 2) + (-1 * 3) = 2 - 3 = -1, thus PQ₃₃ = -1. Voila! We have all the elements of the matrix PQ. Now, let's put it all together.

The Resulting Matrix PQ

Okay, guys, we've done it! We've meticulously calculated all the elements of the matrix PQ. Now, let's put it all together. The resulting matrix PQ is as follows:

$PQ=\left[\begin{array}{ccc} -2 & 0 & 4 \\ 6 & -10 & 23 \\ -3 & 2 & -1 \end{array}\right]$

Congratulations! We've successfully multiplied matrices P and Q. Notice how the resulting matrix is a 3x3 matrix, as we predicted earlier. This serves as a great check to ensure all our calculations are accurate. Matrix multiplication is a fundamental operation, and mastering it will take you a long way in understanding linear algebra and its applications. Practice with different matrices, try variations, and soon, you'll be solving these problems like a pro. Keep practicing and exploring – there's so much more to discover in the world of matrices and linear algebra! I recommend trying some practice problems. Here's a quick tip: when you're first learning, write down the dimensions of each matrix and double-check them before starting. This helps you avoid common mistakes. Also, don't be afraid to break down the calculation into smaller steps, as we did here. This will make the entire process more manageable. With consistent practice, matrix multiplication will become second nature, and you'll be ready to tackle more complex problems. Remember, the journey of a thousand miles begins with a single step. Or, in this case, a single dot product. Go forth and conquer the matrices!