Math Problem Statement
Jika diketahui matriks A = [-2 -1; 5 3], B = [4 3; -6 0], dan C = [-3 -6; -3 -2], maka hasil dari: 2A + BC adalah ...
Solution
The question provides three matrices:
B = \begin{pmatrix} 4 & 3 \\ -6 & 0 \end{pmatrix}, C = \begin{pmatrix} -3 & -6 \\ -3 & -2 \end{pmatrix}$$ The task is to calculate the result of $$2A + BC$$, where: 1. $$2A$$ is the matrix $$A$$ scaled by 2. 2. $$BC$$ is the matrix product of matrices $$B$$ and $$C$$. Let's calculate $$2A$$ and $$BC$$ step by step. ### Step 1: Calculate $$2A$$ $$2A = 2 \times \begin{pmatrix} -2 & -1 \\ 5 & 3 \end{pmatrix} = \begin{pmatrix} -4 & -2 \\ 10 & 6 \end{pmatrix}$$ ### Step 2: Calculate $$BC$$ To calculate the product $$BC$$, we perform matrix multiplication between $$B$$ and $$C$$: $$B = \begin{pmatrix} 4 & 3 \\ -6 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} -3 & -6 \\ -3 & -2 \end{pmatrix}$$ The matrix multiplication $$BC$$ is calculated as follows: $$BC = \begin{pmatrix} (4 \times -3) + (3 \times -3) & (4 \times -6) + (3 \times -2) \\ (-6 \times -3) + (0 \times -3) & (-6 \times -6) + (0 \times -2) \end{pmatrix}$$ $$BC = \begin{pmatrix} -12 + (-9) & -24 + (-6) \\ 18 + 0 & 36 + 0 \end{pmatrix}$$ $$BC = \begin{pmatrix} -21 & -30 \\ 18 & 36 \end{pmatrix}$$ ### Step 3: Calculate $$2A + BC$$ Now, we add the matrices $$2A$$ and $$BC$$: $$2A + BC = \begin{pmatrix} -4 & -2 \\ 10 & 6 \end{pmatrix} + \begin{pmatrix} -21 & -30 \\ 18 & 36 \end{pmatrix}$$ $$2A + BC = \begin{pmatrix} -4 + (-21) & -2 + (-30) \\ 10 + 18 & 6 + 36 \end{pmatrix}$$ $$2A + BC = \begin{pmatrix} -25 & -32 \\ 28 & 42 \end{pmatrix}$$ Thus, the result of $$2A + BC$$ is: $$\begin{pmatrix} -25 & -32 \\ 28 & 42 \end{pmatrix}$$ Would you like more details or have any questions? Here are five related questions: 1. How do you compute the product of two matrices? 2. What is the general formula for multiplying two 2x2 matrices? 3. How do you find the scalar multiple of a matrix? 4. What are the properties of matrix addition? 5. Can matrix multiplication be commutative? **Tip**: Always ensure the number of columns in the first matrix matches the number of rows in the second matrix when performing matrix multiplication!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Scalar Multiplication
Matrix Multiplication
Matrix Addition
Formulas
Scalar Multiplication: kA = k * A
Matrix Multiplication: (AB)_{ij} = Σ(A_{ik} * B_{kj})
Matrix Addition: (A + B)_{ij} = A_{ij} + B_{ij}
Theorems
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Suitable Grade Level
Grades 10-12
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