Math Problem Statement
Find the determinant of matrix A using cofactor expansion or Sarrus' method. The matrix A is:
A = (2, 3, 1; 4, 1, 5; 3, 2, 0).
Solution
The matrix in the image is:
2 & 3 & 1 \\ 4 & 1 & 5 \\ 3 & 2 & 0 \end{pmatrix}$$ It seems like you are tasked with finding the determinant of this matrix using cofactor expansion or Sarrus' rule. ### To find the determinant using Sarrus' rule: For a $$3 \times 3$$ matrix $$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$, the determinant can be found as: $$\text{det}(A) = aei + bfg + cdh - ceg - bdi - afh$$ ### Let's calculate it for the matrix $$A$$: $$A = \begin{pmatrix} 2 & 3 & 1 \\ 4 & 1 & 5 \\ 3 & 2 & 0 \end{pmatrix}$$ Now apply the formula: $$\text{det}(A) = (2 \cdot 1 \cdot 0) + (3 \cdot 5 \cdot 3) + (1 \cdot 4 \cdot 2) - (1 \cdot 1 \cdot 3) - (3 \cdot 4 \cdot 0) - (2 \cdot 5 \cdot 2)$$ Let me compute that for you.The determinant of the matrix $$A$$ is: $$\text{det}(A) = 30$$ Would you like further details or explanations on how the determinant was calculated? Here are 5 related questions to expand your understanding: 1. What is cofactor expansion, and how does it compare to Sarrus' rule? 2. How would you compute the inverse of a $$3 \times 3$$ matrix if the determinant is non-zero? 3. What are some practical applications of determinants in real-world problems? 4. How does Sarrus' rule apply to matrices larger than $$3 \times 3$$? 5. What is the significance of a matrix having a zero determinant? **Tip:** A matrix with a zero determinant is called singular, meaning it doesn't have an inverse.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Determinants
Sarrus' Rule
Cofactor Expansion
Formulas
Determinant of a 3x3 matrix using Sarrus' Rule: det(A) = aei + bfg + cdh - ceg - bdi - afh
Theorems
Sarrus' Rule
Cofactor Expansion
Suitable Grade Level
Grades 10-12