Math Problem Statement
matriks p=x-2y 4 0 2 2x-1 1 5 3 y+3 mempunyai trace 1 nilai 3x-y adalah
Solution
To solve this problem, we first need to understand what a trace of a matrix is. The trace of a matrix is the sum of the elements on its main diagonal (from the top left to the bottom right).
Given the matrix :
x - 2y & 4 \\ 0 & 2 \end{bmatrix} \begin{bmatrix} 2x - 1 & 1 \\ 5 & 3 \end{bmatrix} \begin{bmatrix} y + 3 \end{bmatrix}$$ Let's interpret this as a single 3x3 matrix for calculating its trace: $$P = \begin{bmatrix} x - 2y & 4 & 0 \\ 2 & 2x - 1 & 1 \\ 5 & 3 & y + 3 \end{bmatrix}$$ The trace of $$P$$ is: $$\text{Trace}(P) = (x - 2y) + (2x - 1) + (y + 3)$$ Given that the trace is equal to 1, we set up the equation: $$(x - 2y) + (2x - 1) + (y + 3) = 1$$ Simplifying this equation: $$x - 2y + 2x - 1 + y + 3 = 1$$ $$3x - y + 2 = 1$$ Subtract 2 from both sides: $$3x - y = -1$$ Therefore, the value of $$3x - y$$ is **-1**. ### Additional Questions 1. What are the properties of the trace of a matrix? 2. How do you find the determinant of a matrix? 3. What is the significance of eigenvalues in a matrix? 4. Can a non-square matrix have a trace? Why or why not? 5. How does changing elements outside the main diagonal affect the trace of a matrix? ### Tip Remember, the trace of a matrix is simply the sum of the diagonal elements. This makes it a very easy property to calculate and utilize, especially in linear algebra applications.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Trace of a Matrix
Formulas
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Theorems
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Suitable Grade Level
Advanced High School
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