Math Problem Statement
Cari nilai m dan n bagi persamaan \(\begin{pmatrix} \frac{m}{2} & 10 \\ m + n & n \end{pmatrix} = \begin{pmatrix} -3 & 10 \\ 2 & 8 \end{pmatrix}.\)
Solution
The problem asks us to find the values of and that satisfy the given equation involving matrices:
\frac{m}{2} & 10 \\ m + n & n \end{pmatrix} = \begin{pmatrix} -3 & 10 \\ 2 & 8 \end{pmatrix}.$$ ### Step-by-Step Solution: 1. **Equating the elements of the matrices**: Compare the corresponding elements of the two matrices to form equations: - Top left element: $$\frac{m}{2} = -3$$. - Top right element: $$10 = 10$$ (this is already satisfied). - Bottom left element: $$m + n = 2$$. - Bottom right element: $$n = 8$$. 2. **Solve for $$m$$ and $$n$$**: - From $$\frac{m}{2} = -3$$: $$m = -6.$$ - From $$n = 8$$ (directly given): $$n = 8.$$ - From $$m + n = 2$$: Substituting $$m = -6$$ and $$n = 8$$, $$-6 + 8 = 2,$$ which is correct. ### Final Answer: $$m = -6, \quad n = 8.$$ Let me know if you'd like further clarification or details! --- ### Related Questions: 1. What does it mean for two matrices to be equal? 2. How would the solution change if $$n$$ was not directly given as part of the equation? 3. Can you explain the process of equating matrix elements in more detail? 4. How are matrices applied in real-world problems? 5. How can we verify the solution of this problem? ### Tip: When solving for unknowns in matrix equations, always check each individual element of the matrices to ensure all parts of the equation are satisfied.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Equations
Linear Algebra
Element-by-Element Comparison
Formulas
Matrix element equality
Theorems
Equality of Matrices
Suitable Grade Level
Grades 10-12