Math Problem Statement
Diketahui persamaan matriks berikut, Matriks X adalah?
Solution
To solve the given matrix equation, let's analyze it step by step:
The equation is:
3 & 2 \\ -5 & -3 \end{bmatrix} \cdot X = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ Where $$X$$ is a $$2 \times 2$$ matrix. ### Step 1: Write the equation for $$X$$ To isolate $$X$$, we multiply both sides of the equation by the inverse of the matrix $$\begin{bmatrix} 3 & 2 \\ -5 & -3 \end{bmatrix}$$ on the left. This gives: $$X = \begin{bmatrix} 3 & 2 \\ -5 & -3 \end{bmatrix}^{-1} \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ ### Step 2: Calculate the inverse of $$\begin{bmatrix} 3 & 2 \\ -5 & -3 \end{bmatrix}$$ The formula for the inverse of a $$2 \times 2$$ matrix is: $$\text{If } A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ For $$\begin{bmatrix} 3 & 2 \\ -5 & -3 \end{bmatrix}$$, we have: - $$a = 3$$, $$b = 2$$, $$c = -5$$, $$d = -3$$ - Determinant = $$(3)(-3) - (2)(-5) = -9 + 10 = 1$$ Thus, the inverse is: $$\begin{bmatrix} 3 & 2 \\ -5 & -3 \end{bmatrix}^{-1} = \begin{bmatrix} -3 & -2 \\ 5 & 3 \end{bmatrix}$$ ### Step 3: Perform the matrix multiplication Now, compute: $$X = \begin{bmatrix} -3 & -2 \\ 5 & 3 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ Multiply the matrices: $$X = \begin{bmatrix} (-3)(1) + (-2)(3) & (-3)(2) + (-2)(4) \\ (5)(1) + (3)(3) & (5)(2) + (3)(4) \end{bmatrix}$$ Simplify: $$X = \begin{bmatrix} -3 - 6 & -6 - 8 \\ 5 + 9 & 10 + 12 \end{bmatrix}$$ $$X = \begin{bmatrix} -9 & -14 \\ 14 & 22 \end{bmatrix}$$ ### Final Answer: The matrix $$X$$ is: $$X = \begin{bmatrix} -9 & -14 \\ 14 & 22 \end{bmatrix}$$ Let me know if you want a detailed explanation of any step! #### Related Questions: 1. How do you compute the determinant of a $$2 \times 2$$ matrix? 2. What are the properties of matrix inverses? 3. Can all matrices be inverted? If not, why? 4. How does matrix multiplication work in terms of rows and columns? 5. What are practical applications of solving matrix equations? #### Tip: Always check the determinant before attempting to find the inverse of a matrix. If it's zero, the matrix is not invertibleAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Matrix Inversion
Matrix Multiplication
Formulas
Inverse of 2x2 matrix: A^-1 = (1 / (ad - bc)) * [[d, -b], [-c, a]]
Matrix Multiplication: C = A * B
Theorems
Matrix Inverse Theorem
Properties of Determinants
Suitable Grade Level
Grades 10-12