Math Problem Statement
Solution
To solve the matrix multiplication problem in the image, the calculation follows this structure:
The problem involves multiplying two 3x3 matrices and then subtracting a scalar multiple of another 2x2 matrix.
Step 1: Multiply the two 3x3 matrices:
Matrix A:
1 & 0 & 2 \\ 0 & 4 & -3 \\ -3 & 1 & 1 \end{pmatrix}$$ Matrix B: $$\begin{pmatrix} 5 & 0 & -1 \\ 2 & -1 & 0 \\ 0 & -1 & -1 \end{pmatrix}$$ We need to calculate the product $$A \times B$$. ### Step 2: Compute the product of A and B. #### First row: - For the first element: $$(1 \times 5) + (0 \times 2) + (2 \times 0) = 5$$ - For the second element: $$(1 \times 0) + (0 \times -1) + (2 \times -1) = -2$$ - For the third element: $$(1 \times -1) + (0 \times 0) + (2 \times -1) = -3$$ #### Second row: - For the first element: $$(0 \times 5) + (4 \times 2) + (-3 \times 0) = 8$$ - For the second element: $$(0 \times 0) + (4 \times -1) + (-3 \times -1) = -1$$ - For the third element: $$(0 \times -1) + (4 \times 0) + (-3 \times -1) = 3$$ #### Third row: - For the first element: $$(-3 \times 5) + (1 \times 2) + (1 \times 0) = -13$$ - For the second element: $$(-3 \times 0) + (1 \times -1) + (1 \times -1) = -2$$ - For the third element: $$(-3 \times -1) + (1 \times 0) + (1 \times -1) = 2$$ Thus, the product $$A \times B$$ is: $$\begin{pmatrix} 5 & -2 & -3 \\ 8 & -1 & 3 \\ -13 & -2 & 2 \end{pmatrix}$$ ### Step 3: Subtract 2 times the second 2x2 matrix: Second matrix (given in the image): $$\begin{pmatrix} -3 & 1 \\ 2 & -5 \end{pmatrix}$$ Multiplying this by -2: $$\begin{pmatrix} (-2) \times (-3) & (-2) \times (1) \\ (-2) \times (2) & (-2) \times (-5) \end{pmatrix} = \begin{pmatrix} 6 & -2 \\ -4 & 10 \end{pmatrix}$$ ### Step 4: Subtract this result from the first 2x2 submatrix of $$A \times B$$: First 2x2 submatrix of $$A \times B$$: $$\begin{pmatrix} 5 & -2 \\ 8 & -1 \end{pmatrix}$$ Subtract: $$\begin{pmatrix} 5 & -2 \\ 8 & -1 \end{pmatrix} - \begin{pmatrix} 6 & -2 \\ -4 & 10 \end{pmatrix} = \begin{pmatrix} 5 - 6 & -2 - (-2) \\ 8 - (-4) & -1 - 10 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 12 & -11 \end{pmatrix}$$ ### Final answer: The result is: $$\begin{pmatrix} -1 & 0 \\ 12 & -11 \end{pmatrix}$$ Thus, the correct answer is **C**. --- Would you like a further breakdown of any steps? Here are 5 related questions to enhance your understanding: 1. How do you multiply two matrices together step by step? 2. What are some key properties of matrix multiplication? 3. What is the importance of scalar multiplication in matrices? 4. How do matrix dimensions affect the outcome of multiplication? 5. How does matrix subtraction work in comparison to addition? **Tip:** Always ensure that matrix dimensions align for multiplication, as this operation is only defined when the number of columns in the first matrix matches the number of rows in the second matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix multiplication
Scalar multiplication
Matrix subtraction
Formulas
Matrix product AxB for elements (i,j): sum(a_ik * b_kj for k in range(n))
Scalar multiplication of matrix C by scalar n: nC
Matrix subtraction A-B for elements (i,j): a_ij - b_ij
Theorems
-
Suitable Grade Level
College or advanced high school courses
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