Math Problem Statement
Diketahui matriks A dan B, jika A + B = C, maka nilai p, q, dan r berturut-turut adalah?
Solution
We are given three matrices , , and , with the equation , and asked to determine the values of , , and .
Let's write the matrices out clearly:
2p & 2 & -3a \\ 4 & -1 & -4 \\ r & q & -2 \end{pmatrix}, \quad B = \begin{pmatrix} -p & -7 & q \\ -5 & 4 & r \\ -5 & 4 & 7 \end{pmatrix}, \quad C = \begin{pmatrix} 2 & -5 & 6 \\ -1 & 4 & -2 \\ -3 & 1 & 5 \end{pmatrix}$$ We know that $$A + B = C$$. Therefore, for each element, we have the following system of equations by comparing corresponding elements: ### First row: $$(2p) + (-p) = 2 \quad \Rightarrow \quad p = 2$$ $$2 + (-7) = -5 \quad \text{(This is already correct)}$$ $$-3a + q = 6$$ ### Second row: $$4 + (-5) = -1 \quad \text{(This is correct)}$$ $$-1 + 4 = 4 \quad \text{(This is correct)}$$ $$-4 + r = -2 \quad \Rightarrow \quad r = 2$$ ### Third row: $$r + (-5) = -3 \quad \Rightarrow \quad r = 2 \quad \text{(This is consistent)}$$ $$q + 4 = 1 \quad \Rightarrow \quad q = -3$$ $$-2 + 7 = 5 \quad \text{(This is correct)}$$ Thus, the values are: $$p = 2, \quad q = -3, \quad r = 2$$ The correct answer is option **D**: 2, -3, and 2. Would you like more details or have further questions? Here are some related questions: 1. What is the definition of matrix addition? 2. How do we solve systems of linear equations? 3. How are matrix dimensions important when adding matrices? 4. What happens when matrices have unequal dimensions? 5. How can matrix equations be solved using Gaussian elimination? **Tip:** Always ensure that matrices are of the same dimension when performing addition.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix addition
Systems of linear equations
Algebra
Formulas
A + B = C, where each element in the resulting matrix is the sum of corresponding elements of matrices A and B
Theorems
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Suitable Grade Level
Grades 10-12 or early college-level