Math Problem Statement
Consider the following. C =
2 −4 12
0 6 −12 1 −2 3 and D = 1 6
1 2 4
2 1 −4 1 0 −2 , Find CD.
Find DC.
Solution
We are tasked with finding the matrix products and , where the given matrices and are as follows:
2 & -4 & 12 \\ 0 & 6 & -12 \\ 1 & -2 & 3 \end{bmatrix} \quad \text{and} \quad D = \begin{bmatrix} 1 & 6 & 1 \\ 2 & 4 & 2 \\ 1 & 0 & -2 \end{bmatrix}$$ ### Step 1: Find $$CD$$ (Matrix multiplication $$C \times D$$) Matrix multiplication is performed by taking the dot product of rows of $$C$$ with columns of $$D$$. $$CD = \begin{bmatrix} (2)(1) + (-4)(2) + (12)(1) & (2)(6) + (-4)(4) + (12)(0) & (2)(1) + (-4)(2) + (12)(-2) \\ (0)(1) + (6)(2) + (-12)(1) & (0)(6) + (6)(4) + (-12)(0) & (0)(1) + (6)(2) + (-12)(-2) \\ (1)(1) + (-2)(2) + (3)(1) & (1)(6) + (-2)(4) + (3)(0) & (1)(1) + (-2)(2) + (3)(-2) \end{bmatrix}$$ Now, calculate each element: $$CD = \begin{bmatrix} 2 - 8 + 12 & 12 - 16 + 0 & 2 - 8 - 24 \\ 0 + 12 - 12 & 0 + 24 + 0 & 0 + 12 + 24 \\ 1 - 4 + 3 & 6 - 8 + 0 & 1 - 4 - 6 \end{bmatrix} = \begin{bmatrix} 6 & -4 & -30 \\ 0 & 24 & 36 \\ 0 & -2 & -9 \end{bmatrix}$$ ### Step 2: Find $$DC$$ (Matrix multiplication $$D \times C$$) Now, calculate $$D \times C$$ by multiplying rows of $$D$$ by columns of $$C$$. $$DC = \begin{bmatrix} (1)(2) + (6)(0) + (1)(1) & (1)(-4) + (6)(6) + (1)(-2) & (1)(12) + (6)(-12) + (1)(3) \\ (2)(2) + (4)(0) + (2)(1) & (2)(-4) + (4)(6) + (2)(-2) & (2)(12) + (4)(-12) + (2)(3) \\ (1)(2) + (0)(0) + (-2)(1) & (1)(-4) + (0)(6) + (-2)(-2) & (1)(12) + (0)(-12) + (-2)(3) \end{bmatrix}$$ Now, calculate each element: $$DC = \begin{bmatrix} 2 + 0 + 1 & -4 + 36 - 2 & 12 - 72 + 3 \\ 4 + 0 + 2 & -8 + 24 - 4 & 24 - 48 + 6 \\ 2 + 0 - 2 & -4 + 0 + 4 & 12 + 0 - 6 \end{bmatrix} = \begin{bmatrix} 3 & 30 & -57 \\ 6 & 12 & -18 \\ 0 & 0 & 6 \end{bmatrix}$$ ### Final Answers: $$CD = \begin{bmatrix} 6 & -4 & -30 \\ 0 & 24 & 36 \\ 0 & -2 & -9 \end{bmatrix} \quad \text{and} \quad DC = \begin{bmatrix} 3 & 30 & -57 \\ 6 & 12 & -18 \\ 0 & 0 & 6 \end{bmatrix}$$ Would you like any further clarifications or detailed explanations? --- Here are five related questions to expand your understanding: 1. What are the conditions for matrix multiplication to be valid? 2. How does the size of the resulting matrix in a multiplication depend on the original matrices? 3. What are the key differences between $$AB$$ and $$BA$$ for non-square matrices? 4. How does matrix multiplication relate to linear transformations? 5. How can you interpret matrix multiplication in terms of row and column operations? **Tip:** Matrix multiplication is not commutative, meaning $$AB \neq BA$$ in most cases!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Algebra
Formulas
Matrix multiplication formula (dot product of rows and columns)
Theorems
Associative property of matrix multiplication
Suitable Grade Level
Grades 11-12