Calculating Matrix Operations: Can We Find (3)BA + (4)AC?

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Hey guys! Today, we're diving into the world of matrix operations to figure out if we can actually calculate (3)BA + (4)AC given specific matrices for A, B, and C. This involves checking if the dimensions of the matrices are compatible for multiplication and addition. It might sound a bit intimidating, but don't worry, we'll break it down step by step so it's super clear. Let's get started and see how it all works out!

Understanding Matrix Dimensions

Before we even think about multiplying or adding matrices, the most crucial thing is understanding their dimensions. The dimensions tell us how many rows and columns a matrix has, and this is key to knowing if operations are even possible. We always write dimensions as "rows × columns". So, a 2 × 3 matrix has 2 rows and 3 columns.

Why is this so important? Well, for matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. If they don't match, you can't multiply them. It's like trying to fit puzzle pieces together – they need to have the right shape! For matrix addition (or subtraction), the matrices must have the exact same dimensions. You can only add or subtract matrices that have the same number of rows and the same number of columns. Think of it like adding apples to apples; you can't directly add apples to oranges!

In our specific problem, we have three matrices: A, B, and C. Let's take a closer look at their dimensions to see what we're working with. This will be our first step in figuring out if we can calculate (3)BA + (4)AC. Getting these basics right is super important, so let's make sure we're all on the same page before moving on to the actual calculations.

Dimensions of A, B, and C

Okay, let's get down to the specifics and identify the dimensions of our matrices A, B, and C. This is absolutely crucial because, as we just discussed, the dimensions dictate whether or not we can perform matrix multiplication and addition. We need to know the shape of each matrix before we can start crunching any numbers!

So, here are our matrices again:

A = [[3, -2, -1], [0, 0, -1]]

B = [[-2, 1], [-2, 4]]

C = [[-1, 0, 0], [0, 1, -1], [-3, 1, -2]]

Let's break down each one:

  • Matrix A: If we count the rows and columns, we see that A has 2 rows and 3 columns. So, the dimensions of A are 2 × 3.
  • Matrix B: B has 2 rows and 2 columns, making its dimensions 2 × 2. This is a square matrix, which is a special case where the number of rows equals the number of columns.
  • Matrix C: C has 3 rows and 3 columns. This also makes C a square matrix, specifically a 3 × 3 matrix.

Now that we know the dimensions of A, B, and C, we have the foundational information we need. The next step is to use these dimensions to determine if the matrix multiplications BA and AC are even possible. This is where the real fun begins, as we start to see how these dimensions interact to allow (or disallow) certain operations.

Checking for Compatibility

Alright, now that we know the dimensions of our matrices (A is 2 × 3, B is 2 × 2, and C is 3 × 3), we can dive into the crucial step of checking for compatibility. Remember, matrix multiplication isn't as simple as just lining up any two matrices and multiplying; the dimensions have to play nice together. This is where we see if the number of columns in the first matrix matches the number of rows in the second matrix. If they don't match, it's a no-go for multiplication!

We need to figure out if we can calculate both BA and AC. Let's take them one at a time:

Checking BA

To multiply B and A, we need to see if the number of columns in B is the same as the number of rows in A. B is a 2 × 2 matrix, so it has 2 columns. A is a 2 × 3 matrix, so it has 2 rows. Great! The number of columns in B (which is 2) does match the number of rows in A (which is also 2). This means we can multiply B and A.

But what will the resulting matrix BA look like? The dimensions of the resulting matrix will be the number of rows in the first matrix (B, which has 2 rows) by the number of columns in the second matrix (A, which has 3 columns). So, BA will be a 2 × 3 matrix. This is important to keep in mind for the next steps!

Checking AC

Now let's see if we can multiply A and C. A is a 2 × 3 matrix, so it has 3 columns. C is a 3 × 3 matrix, meaning it has 3 rows. Again, we have a match! The number of columns in A (3) is the same as the number of rows in C (3). So, we can multiply A and C as well.

What about the dimensions of the resulting matrix AC? It will have the same number of rows as A (2 rows) and the same number of columns as C (3 columns). Therefore, AC will also be a 2 × 3 matrix. This is another key piece of information as we move towards calculating (3)BA + (4)AC.

The Big Picture

We've now confirmed that both BA and AC are possible multiplications. This is fantastic news because it means we're on the right track! We also know that both BA and AC will be 2 × 3 matrices. This is especially important because, to add (3)BA and (4)AC, they need to have the same dimensions. And guess what? They do! This means the final addition step is also possible.

So, we've cleared the first hurdle – checking for compatibility. Now, the real work begins: actually performing the matrix multiplications. Buckle up, because we're about to dive into the nitty-gritty calculations!

Calculating BA and AC

Okay, guys, we've confirmed that we can multiply B by A and A by C. Now it's time to roll up our sleeves and actually do the multiplications. This is where we put the definitions of matrix multiplication into action. Remember, it's not just multiplying corresponding elements; it's a bit more involved than that. We're going to take it step by step, so don't worry if it seems a little complex at first. Once you get the hang of it, it's like riding a bike!

Calculating BA

Let's start with BA. We have:

B = [[-2, 1], [-2, 4]]

A = [[3, -2, -1], [0, 0, -1]]

To find the elements of BA, we'll use the following rule: the element in the ith row and jth column of BA is the dot product of the ith row of B and the jth column of A. Sounds complicated? Let's break it down:

  • Element (1,1) of BA: This is the dot product of the first row of B ([-2, 1]) and the first column of A ([3, 0]). So, it's (-2 * 3) + (1 * 0) = -6.
  • Element (1,2) of BA: This is the dot product of the first row of B ([-2, 1]) and the second column of A ([-2, 0]). So, it's (-2 * -2) + (1 * 0) = 4.
  • Element (1,3) of BA: This is the dot product of the first row of B ([-2, 1]) and the third column of A ([-1, -1]). So, it's (-2 * -1) + (1 * -1) = 1.
  • Element (2,1) of BA: This is the dot product of the second row of B ([-2, 4]) and the first column of A ([3, 0]). So, it's (-2 * 3) + (4 * 0) = -6.
  • Element (2,2) of BA: This is the dot product of the second row of B ([-2, 4]) and the second column of A ([-2, 0]). So, it's (-2 * -2) + (4 * 0) = 4.
  • Element (2,3) of BA: This is the dot product of the second row of B ([-2, 4]) and the third column of A ([-1, -1]). So, it's (-2 * -1) + (4 * -1) = -2.

Putting it all together, we get:

BA = [[-6, 4, 1], [-6, 4, -2]]

Calculating AC

Now let's tackle AC. We have:

A = [[3, -2, -1], [0, 0, -1]]

C = [[-1, 0, 0], [0, 1, -1], [-3, 1, -2]]

We'll use the same dot product method as before:

  • Element (1,1) of AC: (3 * -1) + (-2 * 0) + (-1 * -3) = 0
  • Element (1,2) of AC: (3 * 0) + (-2 * 1) + (-1 * 1) = -3
  • Element (1,3) of AC: (3 * 0) + (-2 * -1) + (-1 * -2) = 4
  • Element (2,1) of AC: (0 * -1) + (0 * 0) + (-1 * -3) = 3
  • Element (2,2) of AC: (0 * 0) + (0 * 1) + (-1 * 1) = -1
  • Element (2,3) of AC: (0 * 0) + (0 * -1) + (-1 * -2) = 2

So, we have:

AC = [[0, -3, 4], [3, -1, 2]]

We've done it! We've successfully calculated BA and AC. This was the most computationally intensive part of the problem. Now, we're in the home stretch. The next step is to multiply these resulting matrices by the scalar values (3) and (4), respectively, and then add them together. We're almost there!

Scalar Multiplication and Matrix Addition

Alright, we've made it to the final stretch! We've already calculated BA and AC. Now, we need to multiply each of these matrices by a scalar (a regular number) and then add the resulting matrices together. This is where the (3) and (4) in (3)BA + (4)AC come into play. Scalar multiplication is pretty straightforward, and matrix addition is a breeze once you've got the dimensions right (which we do!). Let's dive in and finish this problem off!

Scalar Multiplication

Scalar multiplication is super simple: you just multiply every element in the matrix by the scalar. It's like distributing the scalar across the entire matrix. So, let's do this for 3BA and 4AC.

We know:

BA = [[-6, 4, 1], [-6, 4, -2]]

AC = [[0, -3, 4], [3, -1, 2]]

First, let's find 3BA:

3BA = 3 * [[-6, 4, 1], [-6, 4, -2]] = [[3 * -6, 3 * 4, 3 * 1], [3 * -6, 3 * 4, 3 * -2]] = [[-18, 12, 3], [-18, 12, -6]]

Next, let's find 4AC:

4AC = 4 * [[0, -3, 4], [3, -1, 2]] = [[4 * 0, 4 * -3, 4 * 4], [4 * 3, 4 * -1, 4 * 2]] = [[0, -12, 16], [12, -4, 8]]

So, we've now calculated 3BA and 4AC. We're just one step away from the final answer!

Matrix Addition

The last step is to add 3BA and 4AC together. Remember, we can only add matrices if they have the same dimensions, which we already confirmed they do (both are 2 × 3 matrices). To add matrices, you simply add the corresponding elements. That means you add the element in the first row and first column of the first matrix to the element in the first row and first column of the second matrix, and so on.

We have:

3BA = [[-18, 12, 3], [-18, 12, -6]]

4AC = [[0, -12, 16], [12, -4, 8]]

So, (3)BA + (4)AC is:

[[-18 + 0, 12 + (-12), 3 + 16], [-18 + 12, 12 + (-4), -6 + 8]] = [[-18, 0, 19], [-6, 8, 2]]

Final Answer

And there you have it! We've successfully calculated (3)BA + (4)AC. It was a journey, but we broke it down step by step, from checking dimensions to performing matrix multiplication, scalar multiplication, and finally, matrix addition. You guys are matrix operation superstars now!

Our final answer is:

(3)BA + (4)AC = [[-18, 0, 19], [-6, 8, 2]]

Key Takeaways:

  • Dimensions are King: Always, always check the dimensions of your matrices before attempting any operations. They dictate what's possible and what's not.
  • Matrix Multiplication: Remember the dot product rule. It might seem tricky at first, but practice makes perfect!
  • Scalar Multiplication: This is the easy part – just multiply every element by the scalar.
  • Matrix Addition: Make sure the matrices have the same dimensions, and then add corresponding elements.

I hope this breakdown was helpful and made matrix operations a little less mysterious. Keep practicing, and you'll be a pro in no time!