Math Problem Statement
Diketahui matriks A = [ [a + 2, 1 - 3b], [-1, -6] ], B = [ [2a, b - 3], [-1, 2] ], dan C = [ [5, 6], [-2, -4] ]. Jika A + B = C, tentukan nilai 3a + b!
Solution
We are given three matrices:
B = \begin{bmatrix} 2a & b - 3 \\ -1 & 2 \end{bmatrix}, \quad C = \begin{bmatrix} 5 & 6 \\ -2 & -4 \end{bmatrix}$$ The problem states that $$A + B = C$$, and we are asked to determine the value of $$3a + b$$. ### Step 1: Matrix Addition Equation Given that $$A + B = C$$, we can set up an element-wise equation by adding matrices $$A$$ and $$B$$: $$A + B = \begin{bmatrix} (a+2) + 2a & (1 - 3b) + (b - 3) \\ -1 + (-1) & -6 + 2 \end{bmatrix} = \begin{bmatrix} 3a + 2 & 1 - 2b - 3 \\ -2 & -4 \end{bmatrix}$$ We know that $$A + B$$ must equal $$C$$: $$\begin{bmatrix} 3a + 2 & 1 - 2b - 3 \\ -2 & -4 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ -2 & -4 \end{bmatrix}$$ ### Step 2: Solve for $$a$$ and $$b$$ By comparing corresponding elements of the matrices: 1. From the element in the first row, first column: $$3a + 2 = 5 \quad \Rightarrow \quad 3a = 3 \quad \Rightarrow \quad a = 1$$ 2. From the element in the first row, second column: $$1 - 2b - 3 = 6 \quad \Rightarrow \quad -2b - 2 = 6 \quad \Rightarrow \quad -2b = 8 \quad \Rightarrow \quad b = -4$$ ### Step 3: Calculate $$3a + b$$ Substitute $$a = 1$$ and $$b = -4$$ into the expression $$3a + b$$: $$3a + b = 3(1) + (-4) = 3 - 4 = -1$$ Thus, the value of $$3a + b$$ is $$-1$$. --- Would you like more details or have any questions? Here are 5 related questions: 1. How do you verify if two matrices are equal? 2. Can matrix addition be performed with matrices of different dimensions? 3. What are the properties of matrix addition? 4. How is matrix multiplication different from matrix addition? 5. What is the geometric interpretation of matrix operations? **Tip:** When comparing matrices, always ensure that corresponding elements are compared one by one to solve for unknowns.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Addition
Linear Algebra
Formulas
Matrix Addition Formula
System of Linear Equations
Theorems
Equality of Matrices
Suitable Grade Level
Grades 10-12