Matrix Multiplication for A = [[6, 4, 3], [1, 0, -1]] and B = [[3, 3], [4, -2], [2, 5]]

Math Problem Statement

Matrix multiplication involving two matrices: A = [[6, 4, 3], [1, 0, -1]] and B = [[3, 3], [4, -2], [2, 5]].

It appears you have two matrices in the image. Let's label them as follows for clarity:

Matrix A:

6 & 4 & 3 \\ 1 & 0 & -1 \end{pmatrix}$$ Matrix B: $$\begin{pmatrix} 3 & 3 \\ 4 & -2 \\ 2 & 5 \end{pmatrix}$$ It seems you may want to multiply these two matrices. To do so, we need to ensure matrix multiplication is valid. The rule for multiplying matrices is that the number of columns in the first matrix (Matrix A) must match the number of rows in the second matrix (Matrix B). Matrix A has dimensions $$2 \times 3$$ (2 rows, 3 columns), and Matrix B has dimensions $$3 \times 2$$ (3 rows, 2 columns), so multiplication is possible. Let's perform the multiplication $$A \times B$$: $$\text{Resulting matrix C} = A \times B$$ The resulting matrix will have dimensions $$2 \times 2$$, because Matrix A has 2 rows and Matrix B has 2 columns. I will now calculate this for you.The result of multiplying the two matrices $$A \times B$$ is: $$\begin{pmatrix} 40 & 25 \\ 1 & -2 \end{pmatrix}$$ Let me know if you need further details or clarification! Here are 5 related questions for further exploration: 1. What is the general rule for matrix multiplication? 2. How do you find the determinant of a $$2 \times 2$$ matrix? 3. Can matrix multiplication be commutative (i.e., is $$A \times B = B \times A$$)? 4. What is the significance of matrix dimensions in linear transformations? 5. How can you find the inverse of a matrix? **Tip:** When multiplying matrices, always verify the dimensions to ensure the operation is valid.

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