Math Problem Statement
) Find the determinant of the following matrix.
A = −3 0 2 0 2 2 0 −3 1 0 −4 −4 1 0 −4 0
Solution
To find the determinant of the given matrix :
-3 & 0 & 2 & 0 \\ 2 & 2 & 0 & -3 \\ 1 & 0 & -4 & -4 \\ 1 & 0 & -4 & 0 \end{bmatrix}$$ ### Step-by-Step Calculation 1. **Choose the first row for expansion:** $$\text{det}(A) = (-3) \cdot C_{11} + 0 \cdot C_{12} + 2 \cdot C_{13} + 0 \cdot C_{14}$$ Here, $$C_{ij}$$ represents the cofactor of the element in the $$i$$-th row and $$j$$-th column. 2. **Calculate the cofactors:** - **Cofactor $$C_{11}$$:** $$C_{11} = \begin{vmatrix} 2 & 0 & -3 \\ 0 & -4 & -4 \\ 0 & -4 & 0 \end{vmatrix}$$ Compute this determinant: $$C_{11} = 2 \cdot ((-4) \cdot 0 - (-4) \cdot (-4)) = 2 \cdot (0 - 16) = -32$$ - **Cofactor $$C_{13}$$:** $$C_{13} = \begin{vmatrix} 2 & 2 & -3 \\ 1 & 0 & -4 \\ 1 & 0 & 0 \end{vmatrix}$$ Compute this determinant: $$C_{13} = 2 \cdot (0 \cdot 0 - (-4) \cdot 0) - 2 \cdot (1 \cdot 0 - (-4) \cdot 1) + (-3) \cdot (1 \cdot 0 - 0 \cdot 1)$$ Simplify: $$C_{13} = 0 + 8 + 0 = 8$$ 3. **Substitute the cofactors back into the determinant expression:** $$\text{det}(A) = (-3)(-32) + 2 \cdot 8$$ Simplify: $$\text{det}(A) = 96 + 16 = 112$$ ### Final Answer: $$\text{det}(A) = 112$$ Would you like more details on the process or have any questions? ### Related Questions 1. How do you compute a determinant for a $$3 \times 3$$ matrix? 2. What is the significance of the determinant being zero? 3. Can you find the inverse of a $$4 \times 4$$ matrix? 4. How do you use cofactor expansion along different rows? 5. What are some applications of determinants in geometry? ### Tip Always choose a row or column with the most zeros for cofactor expansion to simplify the calculations.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Determinants
Cofactor Expansion
Formulas
Determinant of a 4x4 matrix using cofactor expansion
Theorems
Cofactor Expansion Theorem
Suitable Grade Level
Grades 11-12
Related Recommendation
Find the Determinant of a 4x4 Matrix Using Cofactor Expansion
Determinant Calculation of a 4x4 Matrix Using Cofactor Expansion
How to Solve for the Determinant of a 4x4 Matrix Using Cofactor Expansion
Calculate Determinant of a 4x4 Matrix Using Cofactor Expansion
Find the Determinant of a 4x4 Matrix Using Cofactor Expansion