Math Problem Statement
How was the determinant of the matrix calculated?
Solution
The image shows a matrix determinant problem and asks for the method used to calculate the determinant. Here's the breakdown of the solution:
Matrix :
\begin{pmatrix} 1 & 0 & 1 \\ 2 & 3 & 5 \\ 4 & 0 & 8 \end{pmatrix}$$ The determinant of $$A$$ is calculated as: $$|A| = 1 \cdot (3 \cdot 8 - 5 \cdot 0) - 0 \cdot (2 \cdot 8 - 5 \cdot 4) + 1 \cdot (2 \cdot 0 - 3 \cdot 4)$$ This expands to: $$|A| = 1 \cdot (24 - 0) - 0 \cdot (16 - 20) + 1 \cdot (0 - 12)$$ $$|A| = 1 \cdot 24 + 0 + (-12) = 24 - 12 = 12$$ ### Question: How was the determinant of the matrix calculated? ### Options: 1. **Using Laplace's theorem (теоремы Лапласа)** 2. **Using elementary transformations (элементарных преобразований)** 3. **Using the triangle formula (формулы треугольника)** ### Correct Answer: The determinant was calculated using **Laplace's theorem** (теоремы Лапласа), as the calculation involves expanding the determinant along the first row. --- Would you like a detailed explanation of Laplace's theorem or clarification on other methods? Let me know! --- ### Follow-up Questions: 1. What is Laplace's theorem, and how is it used for determinant calculations? 2. How do elementary transformations differ from Laplace's method for calculating determinants? 3. Could the determinant be calculated in other ways? If so, how? 4. What is the triangle formula mentioned in the problem? 5. How do you expand a determinant along other rows or columns? ### Tip: When calculating a determinant, always check if simplifying row or column operations can make the process faster!Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Determinants
Formulas
|A| = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Laplace's Theorem
Suitable Grade Level
Undergraduate/Advanced High School (Grades 10-12)