Math Problem Statement
solve
Solution
The image appears to include a matrix operation and a question involving the calculation of the determinant of a 3x3 matrix. Below is a detailed solution to the problem:
Matrix Determinant Calculation
Given the matrix :
1 & 4 & 3 \\ -1 & 5 & 2 \\ 2 & 1 & 1 \\ \end{pmatrix}$$ We need to find the determinant of this matrix, denoted as $$\det(A)$$. ### Step-by-Step Solution: 1. **Expand along the first row:** $$\det(A) = 1 \cdot \det \begin{pmatrix} 5 & 2 \\ 1 & 1 \end{pmatrix} - 4 \cdot \det \begin{pmatrix} -1 & 2 \\ 2 & 1 \end{pmatrix} + 3 \cdot \det \begin{pmatrix} -1 & 5 \\ 2 & 1 \end{pmatrix}$$ 2. **Calculate the determinants of the 2x2 matrices:** - For $$\det \begin{pmatrix} 5 & 2 \\ 1 & 1 \end{pmatrix}$$: $$\det \begin{pmatrix} 5 & 2 \\ 1 & 1 \end{pmatrix} = (5 \cdot 1) - (2 \cdot 1) = 5 - 2 = 3$$ - For $$\det \begin{pmatrix} -1 & 2 \\ 2 & 1 \end{pmatrix}$$: $$\det \begin{pmatrix} -1 & 2 \\ 2 & 1 \end{pmatrix} = (-1 \cdot 1) - (2 \cdot 2) = -1 - 4 = -5$$ - For $$\det \begin{pmatrix} -1 & 5 \\ 2 & 1 \end{pmatrix}$$: $$\det \begin{pmatrix} -1 & 5 \\ 2 & 1 \end{pmatrix} = (-1 \cdot 1) - (5 \cdot 2) = -1 - 10 = -11$$ 3. **Substitute these values back into the determinant formula:** $$\det(A) = 1 \cdot 3 - 4 \cdot (-5) + 3 \cdot (-11)$$ 4. **Simplify the expression:** $$\det(A) = 3 + 20 - 33 = -10$$ So, the determinant of the matrix $$A$$ is $$\boxed{-10}$$. Would you like more details on any part of this solution, or do you have any questions? ### Related Questions: 1. What is the geometric interpretation of the determinant of a matrix? 2. How does the determinant of a matrix change when two rows are swapped? 3. What is the determinant of a triangular matrix? 4. How is the determinant affected by scaling one row of a matrix? 5. Can the determinant be used to determine if a matrix is invertible? 6. How would you find the determinant of a 4x4 matrix? 7. What is the cofactor expansion method for calculating determinants? 8. How does row reduction affect the calculation of the determinant? ### Tip: When calculating determinants, it's often helpful to choose a row or column with the most zeros to simplify the computation.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Operations
Determinants
Formulas
Determinant of a 3x3 matrix
Theorems
-
Suitable Grade Level
Advanced High School