Math Problem Statement
Evaluate det(A) by a cofactor expansion along a row or column of your choice. Matrix A = [[-5, 0, 5], [5, 7, 1], [-1, 0, 6]]
Solution
We are asked to find the determinant of matrix using cofactor expansion along any row or column. The matrix is:
-5 & 0 & 5 \\ 5 & 7 & 1 \\ -1 & 0 & 6 \end{bmatrix}$$ I will perform the cofactor expansion along the first row, as it contains zeros, which simplifies the computation. The formula for the cofactor expansion along the first row is: $$\text{det}(A) = (-5) \cdot \text{det}\begin{bmatrix} 7 & 1 \\ 0 & 6 \end{bmatrix} - 0 \cdot \text{det} \begin{bmatrix} 5 & 1 \\ -1 & 6 \end{bmatrix} + 5 \cdot \text{det} \begin{bmatrix} 5 & 7 \\ -1 & 0 \end{bmatrix}$$ ### Step 1: Calculate $$\text{det} \begin{bmatrix} 7 & 1 \\ 0 & 6 \end{bmatrix}$$ $$\text{det} \begin{bmatrix} 7 & 1 \\ 0 & 6 \end{bmatrix} = (7 \cdot 6) - (1 \cdot 0) = 42$$ ### Step 2: Calculate $$\text{det} \begin{bmatrix} 5 & 7 \\ -1 & 0 \end{bmatrix}$$ $$\text{det} \begin{bmatrix} 5 & 7 \\ -1 & 0 \end{bmatrix} = (5 \cdot 0) - (7 \cdot -1) = 7$$ ### Step 3: Substitute the values into the cofactor expansion formula: $$\text{det}(A) = (-5)(42) + 0 + 5(7)$$ $$\text{det}(A) = -210 + 0 + 35 = -175$$ Thus, the determinant of the matrix $$A$$ is: $$\boxed{-175}$$ Would you like more details or have any questions? Here are five related questions you might consider: 1. How is the cofactor expansion method applied to a 4x4 matrix? 2. What is the significance of choosing a row or column with many zeros for cofactor expansion? 3. Can the cofactor expansion method be used for matrices larger than 3x3? 4. How do you find the determinant of a 2x2 matrix? 5. What is the geometric interpretation of a matrix's determinant? **Tip:** When using cofactor expansion, it's always a good idea to pick a row or column with as many zeros as possible to reduce calculation complexity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Determinants
Cofactor Expansion
Formulas
Cofactor expansion for determinant: det(A) = a11 * det(M11) - a12 * det(M12) + a13 * det(M13)
Determinant of a 2x2 matrix: det([[a, b], [c, d]]) = ad - bc
Theorems
Cofactor Expansion Theorem
Suitable Grade Level
Undergraduate (Math or Engineering students)