Math Problem Statement
Solve BC matrix and show steps plz
Solution
To solve for the matrix product , we'll multiply the matrices and as given in the image. The matrices are:
1 & -3 & 0 \\ -3 & 1 & 0 \\ 0 & 0 & -2 \end{bmatrix}, \quad C = \begin{bmatrix} 0 & 1 \\ 3 & 2 \\ -2 & 0 \end{bmatrix}$$ ### Step 1: Confirm the dimensions - $$B$$ is a $$3 \times 3$$ matrix. - $$C$$ is a $$3 \times 2$$ matrix. Since the number of columns in $$B$$ matches the number of rows in $$C$$, the product $$BC$$ is defined and will result in a $$3 \times 2$$ matrix. ### Step 2: Perform the matrix multiplication To find the element in the $$i$$-th row and $$j$$-th column of $$BC$$, denoted as $$(BC)_{ij}$$, compute the dot product of the $$i$$-th row of $$B$$ with the $$j$$-th column of $$C$$. $$BC = \begin{bmatrix} (BC)_{11} & (BC)_{12} \\ (BC)_{21} & (BC)_{22} \\ (BC)_{31} & (BC)_{32} \end{bmatrix}$$ Where: $$(BC)_{11} = (1)(0) + (-3)(3) + (0)(-2) = 0 - 9 + 0 = -9$$ $$(BC)_{12} = (1)(1) + (-3)(2) + (0)(0) = 1 - 6 + 0 = -5$$ $$(BC)_{21} = (-3)(0) + (1)(3) + (0)(-2) = 0 + 3 + 0 = 3$$ $$(BC)_{22} = (-3)(1) + (1)(2) + (0)(0) = -3 + 2 + 0 = -1$$ $$(BC)_{31} = (0)(0) + (0)(3) + (-2)(-2) = 0 + 0 + 4 = 4$$ $$(BC)_{32} = (0)(1) + (0)(2) + (-2)(0) = 0 + 0 + 0 = 0$$ Thus, the product $$BC$$ is: $$BC = \begin{bmatrix} -9 & -5 \\ 3 & -1 \\ 4 & 0 \end{bmatrix}$$ ### Final Answer: The matrix product $$BC$$ is: $$BC = \begin{bmatrix} -9 & -5 \\ 3 & -1 \\ 4 & 0 \end{bmatrix}$$ Would you like further details or have any questions? Here are 5 related questions: 1. How would the result change if the matrix $$C$$ were transposed before multiplying by $$B$$? 2. What is the determinant of matrix $$B$$? 3. How would you verify the calculation by breaking down each step further? 4. What is the rank of the resulting matrix $$BC$$? 5. How can the inverse of matrix $$B$$, if it exists, be used to solve a system of equations? **Tip:** Always check the dimensions of matrices before performing multiplication to ensure the operation is valid.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Formulas
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Theorems
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Suitable Grade Level
Advanced High School