Math Problem Statement
Given the following matrices. [ G ]
9 -1 0 6 -3 5 3 1 2 -8 10 -2 and [ C ]
10 9 -9 0 -2 1 -7 2 -5 6 8 -1 -4 -6 5 -10
Find [ G ] ⋅ [ C ] T
Solution
To solve the problem of finding the product of matrix and the transpose of matrix , denoted as , follow these steps:
1. Write Down the Matrices
Given matrices and :
\begin{bmatrix} 9 & -1 & 0 & 6 \\ -3 & 5 & 3 & 1 \\ 2 & -8 & 10 & -2 \end{bmatrix}$$ $$C = \begin{bmatrix} 10 & 9 & -9 & 0 \\ -2 & 1 & -7 & 2 \\ -5 & 6 & 8 & -1 \\ -4 & -6 & 5 & -10 \end{bmatrix}$$ ### 2. Find the Transpose of Matrix $$C$$ The transpose of matrix $$C$$, denoted $$C^T$$, is obtained by swapping the rows and columns of $$C$$: $$C^T = \begin{bmatrix} 10 & -2 & -5 & -4 \\ 9 & 1 & 6 & -6 \\ -9 & -7 & 8 & 5 \\ 0 & 2 & -1 & -10 \end{bmatrix}$$ ### 3. Multiply Matrix $$G$$ by Matrix $$C^T$$ To find $$G \cdot C^T$$, multiply each row of $$G$$ by each column of $$C^T$$. - **First Row of $$G$$ with all Columns of $$C^T$$:** $$\text{First element} = (9)(10) + (-1)(9) + (0)(-9) + (6)(0) = 90 - 9 + 0 + 0 = 81$$ $$\text{Second element} = (9)(-2) + (-1)(1) + (0)(-7) + (6)(2) = -18 - 1 + 0 + 12 = -7$$ $$\text{Third element} = (9)(-5) + (-1)(6) + (0)(8) + (6)(-1) = -45 - 6 + 0 - 6 = -57$$ $$\text{Fourth element} = (9)(-4) + (-1)(-6) + (0)(5) + (6)(-10) = -36 + 6 + 0 - 60 = -90$$ - **Second Row of $$G$$ with all Columns of $$C^T$$:** $$\text{First element} = (-3)(10) + (5)(9) + (3)(-9) + (1)(0) = -30 + 45 - 27 + 0 = -12$$ $$\text{Second element} = (-3)(-2) + (5)(1) + (3)(-7) + (1)(2) = 6 + 5 - 21 + 2 = -8$$ $$\text{Third element} = (-3)(-5) + (5)(6) + (3)(8) + (1)(-1) = 15 + 30 + 24 - 1 = 68$$ $$\text{Fourth element} = (-3)(-4) + (5)(-6) + (3)(5) + (1)(-10) = 12 - 30 + 15 - 10 = -13$$ - **Third Row of $$G$$ with all Columns of $$C^T$$:** $$\text{First element} = (2)(10) + (-8)(9) + (10)(-9) + (-2)(0) = 20 - 72 - 90 + 0 = -142$$ $$\text{Second element} = (2)(-2) + (-8)(1) + (10)(-7) + (-2)(2) = -4 - 8 - 70 - 4 = -86$$ $$\text{Third element} = (2)(-5) + (-8)(6) + (10)(8) + (-2)(-1) = -10 - 48 + 80 + 2 = 24$$ $$\text{Fourth element} = (2)(-4) + (-8)(-6) + (10)(5) + (-2)(-10) = -8 + 48 + 50 + 20 = 110$$ ### 4. Write Down the Resulting Matrix The resulting matrix $$G \cdot C^T$$ is: $$G \cdot C^T = \begin{bmatrix} 81 & -7 & -57 & -90 \\ -12 & -8 & 68 & -13 \\ -142 & -86 & 24 & 110 \end{bmatrix}$$ ### Conclusion So, the product of matrix $$G$$ and the transpose of matrix $$C$$ is: $$\begin{bmatrix} 81 & -7 & -57 & -90 \\ -12 & -8 & 68 & -13 \\ -142 & -86 & 24 & 110 \end{bmatrix}$$ Would you like more details, or do you have any questions about this process? --- Here are some related questions to consider: 1. What is matrix multiplication, and how does it work? 2. How do you find the transpose of a matrix? 3. What are the properties of matrix multiplication? 4. Can matrix multiplication be commutative? 5. How do you verify the correctness of a matrix multiplication? 6. What are the applications of matrix multiplication in real-world problems? 7. How does the dimension of matrices affect the multiplication process? 8. How can matrix operations be implemented in programming languages like Python? **Tip:** Always check the dimensions of the matrices before multiplying them to ensure that the operation is valid.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Transpose of a Matrix
Formulas
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Theorems
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Suitable Grade Level
Advanced High School or College
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