Math Problem Statement
Diketahui A = [[3, -4], [1, 2]], B = [[-3, -2], [-1, 5]] dan C = [[5, 4], [-2, -1]]. Tentukan 3A^T - B + 2C !
Solution
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Problem 1:
Given matrices:
B = \begin{bmatrix} -3 & -2 \\ -1 & 5 \end{bmatrix}, C = \begin{bmatrix} 5 & 4 \\ -2 & -1 \end{bmatrix}$$ You are asked to find the value of $$3A^T - B + 2C$$. ### Step-by-step solution: 1. **Transpose of matrix A, $$A^T$$:** Transpose of matrix $$A$$: $$A^T = \begin{bmatrix} 3 & 1 \\ -4 & 2 \end{bmatrix}$$ 2. **Multiply $$A^T$$ by 3:** $$3A^T = 3 \times \begin{bmatrix} 3 & 1 \\ -4 & 2 \end{bmatrix} = \begin{bmatrix} 9 & 3 \\ -12 & 6 \end{bmatrix}$$ 3. **Multiply matrix $$C$$ by 2:** $$2C = 2 \times \begin{bmatrix} 5 & 4 \\ -2 & -1 \end{bmatrix} = \begin{bmatrix} 10 & 8 \\ -4 & -2 \end{bmatrix}$$ 4. **Add and subtract the matrices:** $$3A^T - B + 2C = \begin{bmatrix} 9 & 3 \\ -12 & 6 \end{bmatrix} - \begin{bmatrix} -3 & -2 \\ -1 & 5 \end{bmatrix} + \begin{bmatrix} 10 & 8 \\ -4 & -2 \end{bmatrix}$$ First, perform the subtraction $$3A^T - B$$: $$3A^T - B = \begin{bmatrix} 9 & 3 \\ -12 & 6 \end{bmatrix} - \begin{bmatrix} -3 & -2 \\ -1 & 5 \end{bmatrix} = \begin{bmatrix} 12 & 5 \\ -11 & 1 \end{bmatrix}$$ Now add $$2C$$: $$\begin{bmatrix} 12 & 5 \\ -11 & 1 \end{bmatrix} + \begin{bmatrix} 10 & 8 \\ -4 & -2 \end{bmatrix} = \begin{bmatrix} 22 & 13 \\ -15 & -1 \end{bmatrix}$$ ### Final Answer: The result of $$3A^T - B + 2C$$ is: $$\begin{bmatrix} 22 & 13 \\ -15 & -1 \end{bmatrix}$$ --- Would you like to proceed with another problem from the image or need more details on this one? Here are some related questions: 1. What is the general process of finding the transpose of a matrix? 2. Can you explain how matrix addition and subtraction work? 3. How is scalar multiplication applied to a matrix? 4. What happens if we transpose a product of two matrices? 5. How do we check if a matrix is invertible? ### Tip: When adding and subtracting matrices, always ensure that the dimensions of the matrices are compatible for the operation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Transpose of a Matrix
Matrix Addition
Scalar Multiplication
Formulas
Transpose of a matrix A: A^T
Matrix addition: A + B
Scalar multiplication: kA
Theorems
Basic Matrix Operations
Suitable Grade Level
Grades 10-12