Math Problem Statement
Evaluate the determinant of the given 3x3 matrix by expanding using cofactors.
Solution
The matrix in the image is:
-4 & 2 & 4 \\ 1 & 4 & -2 \\ -5 & -4 & 1 \end{bmatrix}$$ We will compute the determinant of this $$3 \times 3$$ matrix by expanding along the first row using cofactor expansion. The formula for a determinant using cofactor expansion along the first row is: $$\text{det}(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$ where $$C_{ij} = (-1)^{i+j} M_{ij}$$ and $$M_{ij}$$ is the minor determinant after removing the $$i$$-th row and $$j$$-th column. ### Step-by-Step Solution: #### Step 1: Expand along the first row $$\text{det} = (-4) \cdot \begin{vmatrix} 4 & -2 \\ -4 & 1 \end{vmatrix} - 2 \cdot \begin{vmatrix} 1 & -2 \\ -5 & 1 \end{vmatrix} + 4 \cdot \begin{vmatrix} 1 & 4 \\ -5 & -4 \end{vmatrix}$$ #### Step 2: Compute each minor determinant 1. Compute $$\begin{vmatrix} 4 & -2 \\ -4 & 1 \end{vmatrix}$$: $$(4)(1) - (-2)(-4) = 4 - 8 = -4$$ 2. Compute $$\begin{vmatrix} 1 & -2 \\ -5 & 1 \end{vmatrix}$$: $$(1)(1) - (-2)(-5) = 1 - 10 = -9$$ 3. Compute $$\begin{vmatrix} 1 & 4 \\ -5 & -4 \end{vmatrix}$$: $$(1)(-4) - (4)(-5) = -4 + 20 = 16$$ #### Step 3: Substitute into the formula $$\text{det} = (-4)(-4) - 2(-9) + 4(16)$$ #### Step 4: Simplify $$\text{det} = 16 + 18 + 64 = 98$$ ### Final Answer: The determinant of the matrix is: $$\boxed{98}$$ Do you want further clarification or have any questions? --- ### Related Questions: 1. What is the definition of a determinant and its significance? 2. How do we compute a determinant for a $$4 \times 4$$ matrix? 3. What are the properties of determinants that simplify their computation? 4. How can determinants help in solving systems of linear equations? 5. What is the relationship between determinants and matrix inverses? ### Tip: When expanding by cofactors, choose the row or column with the most zeros to minimize calculations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Determinant
Cofactor Expansion
Formulas
det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}
C_{ij} = (-1)^{i+j} M_{ij}
Theorems
Cofactor Expansion Theorem
Suitable Grade Level
Grade 11-12