Math Problem Statement
Use the arrow technique to evaluate the determinant of the matrix:
[-1, 2, 3]
[4, 0, -4]
[1, 5, 2]
Solution
I see an image showing a matrix with the task to evaluate its determinant using the arrow technique. The matrix given is:
-1 & 2 & 3 \\ 4 & 0 & -4 \\ 1 & 5 & 2 \end{matrix} \right)$$ Let's calculate the determinant of this 3x3 matrix using the cofactor expansion method. The determinant of a 3x3 matrix $$\text{det}(A)$$ is calculated as: $$\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)$$ For the matrix: $$A = \left( \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix} \right)$$ Substituting the values from your matrix into the formula: $$a = -1, b = 2, c = 3$$ $$d = 4, e = 0, f = -4$$ $$g = 1, h = 5, i = 2$$ I'll calculate the determinant now.The determinant of the matrix is: $$\text{det} = 16$$ Would you like a more detailed explanation of the steps, or do you have any other questions? Here are some related questions to further explore this topic: 1. What is the cofactor expansion method used in larger matrices? 2. How does the arrow method differ from other determinant calculation techniques? 3. Can the determinant of a matrix be used to solve systems of linear equations? 4. What does the determinant tell us about the invertibility of a matrix? 5. How can you interpret the geometric meaning of a determinant in 3D space? **Tip:** When dealing with large matrices, applying properties like row reduction can simplify the determinant calculation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Determinants
Matrix Algebra
Formulas
det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Cofactor expansion
Suitable Grade Level
Grades 10-12