Matrix Multiplication: A Step-by-Step Guide
Hey guys! Let's dive into a cool math problem involving matrix multiplication. Specifically, we're going to find the product of a 1x3 matrix and a 3x3 matrix. Don't worry, it's easier than it sounds! We'll break it down step by step. The goal is to understand how to perform matrix multiplication and arrive at the correct answer. Matrix multiplication is a fundamental concept in linear algebra, and it has tons of applications in computer graphics, physics, and data science. Once you get the hang of it, you will find it pretty fun and interesting, too.
Understanding the Problem: Matrix Multiplication Basics
Okay, so the problem is: . What we have here is a 1x3 matrix (a matrix with one row and three columns) being multiplied by a 3x3 matrix (a matrix with three rows and three columns). Remember, the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible. In our case, the number of columns in the first matrix (3) does match the number of rows in the second matrix (3), so we're good to go! The result of this multiplication will be a 1x3 matrix. The values a
, b
, and c
are what we need to figure out. This is where the real fun begins.
Matrix multiplication isn't like regular multiplication where you just multiply corresponding numbers. Instead, we take the dot product of each row of the first matrix with each column of the second matrix. The dot product means we multiply corresponding elements and then add the results. Let me show you what I mean. The dot product is super important to grasp, so make sure you understand each step! The result of our calculation will be a single row matrix that looks like this . Let's calculate each element of the resulting matrix. The first element of the resulting matrix (that is, a
) is found by taking the dot product of the first row of the first matrix and the first column of the second matrix . The second element (that is, b
) is found by taking the dot product of the first row of the first matrix and the second column of the second matrix . Lastly, the third element (that is, c
) is found by taking the dot product of the first row of the first matrix and the third column of the second matrix . Now, let's break it down to find the values of a, b, and c.
Step-by-Step Calculation: Finding 'a'
Alright, let's find the value of a
. As we discussed, a
is the result of the dot product of the first row of the first matrix and the first column of the second matrix . Here's how we do it: we multiply the corresponding elements and sum them up: (2 * 1) + (1 * -1) + (0 * 0). Let's go through it: 2 * 1 = 2, 1 * -1 = -1, and 0 * 0 = 0. Now add those together: 2 + (-1) + 0 = 1. Therefore, a
= 1. Easy peasy, right? This step is the foundation of matrix multiplication. Always remember to multiply corresponding elements and then add the results to obtain the dot product. This is a common operation in linear algebra and is also used to calculate other values. Practice makes perfect, so make sure you keep practicing until you get the hang of it. With a little practice, you will get really good at matrix multiplication. Keep in mind that this dot product concept is repeated throughout the whole process. Next, we will calculate the value of b
.
Step-by-Step Calculation: Finding 'b'
Now, let's find the value of b
. We need to take the dot product of the first row of the first matrix and the second column of the second matrix . Again, we multiply the corresponding elements and add the results: (2 * -1) + (1 * -2) + (0 * 1). Let's go: 2 * -1 = -2, 1 * -2 = -2, and 0 * 1 = 0. Adding those together: -2 + (-2) + 0 = -4. So, b
= -4. Notice how we're consistently applying the same dot product rule? That's the key to matrix multiplication. The value of 'b' is -4, which comes from the dot product of the first row of the first matrix and the second column of the second matrix. The dot product is a fundamental operation in linear algebra, and understanding it will help you with other linear algebra problems. We're making progress! We have successfully calculated both a
and b
, and we're on our way to finding the value of c
. Remember, this step is just as important as the previous one. Do not skip any steps. You need to understand each of the steps to understand matrix multiplication.
Step-by-Step Calculation: Finding 'c'
Finally, let's calculate c
. For this, we'll take the dot product of the first row of the first matrix and the third column of the second matrix . The calculation goes like this: (2 * 2) + (1 * 1) + (0 * 1). Let's break it down: 2 * 2 = 4, 1 * 1 = 1, and 0 * 1 = 0. Adding those together: 4 + 1 + 0 = 5. Therefore, c
= 5. So, we have found all three values: a
= 1, b
= -4, and c
= 5. Congrats, guys! You've successfully multiplied the matrices! Finding the value of c
completes the process, and now we have the complete solution. Now, we have . That's the final matrix! Matrix multiplication can seem tricky at first, but with practice, you'll master it. It is important to be very careful when calculating the values. Take your time and do not rush! Double-check your work and you will get the hang of it! The value of c
is 5, which completes our matrix multiplication. Let's summarize our findings.
The Solution: Putting It All Together
So, after all the calculations, we found that: . The resulting matrix is . We found the values of a
, b
, and c
to be 1, -4, and 5, respectively. That means a
= 1, b
= -4, and c
= 5. Remember the rules: the dimensions have to match up for the multiplication to work, and you need to use the dot product. This approach can be expanded to matrices of different sizes. Matrix multiplication is a core concept in linear algebra. Remember, that the number of columns in the first matrix must equal the number of rows in the second matrix. You did a great job. With practice, you'll become even more proficient! I hope this guide helped you understand matrix multiplication better. Keep practicing, and you'll be a matrix multiplication pro in no time. It is important to note that matrix multiplication is not commutative, meaning that A * B is not always equal to B * A. This is important to keep in mind while doing these kinds of calculations. Thanks for sticking around, and happy calculating!