https://file.aiquickdraw.com/mathbot/file/1727207731599zwn1g1bz.jpg

The image you provided contains two matrices side by side, which looks like you're trying to multiply them. The matrices are:

\[
A = \begin{pmatrix} 6 & 4 & 3 \\ 1 & 0 & -1 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 3 \\ 4 & -2 \\ 2 & 5 \end{pmatrix}
\]

Matrix \( A \) is a \( 2 \times 3 \) matrix, and matrix \( B \) is a \( 3 \times 2 \) matrix. To multiply them, the number of columns in \( A \) (which is 3) must equal the number of rows in \( B \) (which is also 3). So, the multiplication is valid, and the resulting matrix will have dimensions \( 2 \times 2 \).

Let's perform the matrix multiplication:

\[
C = A \times B = \begin{pmatrix} 6 & 4 & 3 \\ 1 & 0 & -1 \end{pmatrix} \times \begin{pmatrix} 3 & 3 \\ 4 & -2 \\ 2 & 5 \end{pmatrix}
\]

The resulting matrix \( C \) will have elements calculated as follows:

1. \( C_{11} = (6 \times 3) + (4 \times 4) + (3 \times 2) = 18 + 16 + 6 = 40 \)
2. \( C_{12} = (6 \times 3) + (4 \times -2) + (3 \times 5) = 18 - 8 + 15 = 25 \)
3. \( C_{21} = (1 \times 3) + (0 \times 4) + (-1 \times 2) = 3 + 0 - 2 = 1 \)
4. \( C_{22} = (1 \times 3) + (0 \times -2) + (-1 \times 5) = 3 + 0 - 5 = -2 \)

Thus, the product matrix \( C \) is:

\[
C = \begin{pmatrix} 40 & 25 \\ 1 & -2 \end{pmatrix}
\]

Would you like further details or have any questions?

Here are five related questions to expand on this:
1. How do you determine the dimensions of the resulting matrix after multiplication?
2. What happens if the number of columns in the first matrix doesn't match the number of rows in the second?
3. Can you apply matrix multiplication with two matrices of the same size?
4. How would you calculate the transpose of matrix \( C \)?
5. How do matrix multiplications relate to systems of linear equations?

**Tip:** When multiplying matrices, always ensure that the dimensions align: the number of columns of the first matrix must equal the number of rows of the second!

Multiply the two matrices: A = [[6, 4, 3], [1, 0, -1]] and B = [[3, 3], [4, -2], [2, 5]]

Learn how to multiply the matrices A = [[6, 4, 3], [1, 0, -1]] and B = [[3, 3], [4, -2], [2, 5]]. This step-by-step solution explains the matrix multiplication process and results in a 2x2 matrix.

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