Math Problem Statement
Matrik [[4, 1], [3, 1]] *( matrix -1&&a\ 2a+b&7 matrix )=( matrix 5&5\ 7&-6 matrix ) maka nilai a =
Solution
To solve this problem, we need to determine the value of by equating the product of the given matrices to the result matrix.
The matrices are:
4 & 1 \\ 3 & 1 \end{bmatrix} \cdot \begin{bmatrix} -1 & a \\ 2a + b & 7 \end{bmatrix} = \begin{bmatrix} 5 & 5 \\ 7 & -6 \end{bmatrix}.$$ ### Step 1: Matrix multiplication The general formula for multiplying two $$2 \times 2$$ matrices $$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$ and $$B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$$ is: $$A \cdot B = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}.$$ Applying this to the given matrices: $$\begin{bmatrix} 4 & 1 \\ 3 & 1 \end{bmatrix} \cdot \begin{bmatrix} -1 & a \\ 2a + b & 7 \end{bmatrix} = \begin{bmatrix} 4(-1) + 1(2a + b) & 4a + 1(7) \\ 3(-1) + 1(2a + b) & 3a + 1(7) \end{bmatrix}.$$ Simplify each element: $$\begin{bmatrix} -4 + 2a + b & 4a + 7 \\ -3 + 2a + b & 3a + 7 \end{bmatrix}.$$ ### Step 2: Set up equations by equating with $$\begin{bmatrix} 5 & 5 \\ 7 & -6 \end{bmatrix}$$ By equating corresponding elements, we get: 1. For the top-left element: $$-4 + 2a + b = 5 \quad \Rightarrow \quad 2a + b = 9.$$ 2. For the top-right element: $$4a + 7 = 5 \quad \Rightarrow \quad 4a = -2 \quad \Rightarrow \quad a = -\frac{1}{2}.$$ 3. For the bottom-left element: $$-3 + 2a + b = 7 \quad \Rightarrow \quad 2a + b = 10.$$ 4. For the bottom-right element: $$3a + 7 = -6 \quad \Rightarrow \quad 3a = -13 \quad \Rightarrow \quad a = -\frac{13}{3}.$$ ### Step 3: Resolve $$a$$ From equations (2) and (4), we notice there is a conflict in the value of $$a$$. This suggests there might be a typo or missing clarification in the question. Please verify and clarify. Would you like me to proceed further or recheck any part of the problem?Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Algebra
Systems of Equations
Formulas
Matrix multiplication formula: A·B = [a11*b11 + a12*b21, a11*b12 + a12*b22; a21*b11 + a22*b21, a21*b12 + a22*b22]
Theorems
Matrix multiplication theorem
Suitable Grade Level
Grades 9-12