Math Problem Statement

Calculate the determinant of the matrix \( A = \begin{pmatrix} 1 & 2 & -4 \\ 5 & 0 & 5 \\ 4 & 3 & 2 \end{pmatrix} \).

Solution

The task is to calculate the determinant of the given 3×33 \times 3 matrix AA:

1 & 2 & -4 \\ 5 & 0 & 5 \\ 4 & 3 & 2 \end{pmatrix}$$ ### Determinant of a $$3 \times 3$$ matrix The determinant is calculated using the formula: $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where the elements of the matrix are: $$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ ### Step-by-step solution: 1. **Substitute the matrix elements**: $$\begin{aligned} a &= 1, & b &= 2, & c &= -4, \\ d &= 5, & e &= 0, & f &= 5, \\ g &= 4, & h &= 3, & i &= 2. \end{aligned}$$ 2. **Expand the determinant**: $$\text{det}(A) = 1 \cdot (0 \cdot 2 - 5 \cdot 3) - 2 \cdot (5 \cdot 2 - 5 \cdot 4) + (-4) \cdot (5 \cdot 3 - 0 \cdot 4)$$ 3. **Simplify each term**: - $$1 \cdot (0 \cdot 2 - 5 \cdot 3) = 1 \cdot (0 - 15) = -15$$, - $$-2 \cdot (5 \cdot 2 - 5 \cdot 4) = -2 \cdot (10 - 20) = -2 \cdot (-10) = 20$$, - $$-4 \cdot (5 \cdot 3 - 0 \cdot 4) = -4 \cdot (15 - 0) = -4 \cdot 15 = -60$$. 4. **Add the results**: $$\text{det}(A) = -15 + 20 - 60 = -55.$$ ### Final Answer: $$\boxed{-55}$$ Let me know if you'd like a detailed explanation of any step, or if you have more questions! --- Here are 5 related questions: 1. What is the determinant formula for a $$2 \times 2$$ matrix? 2. How do determinants relate to matrix inverses? 3. How can the determinant be used to identify singular matrices? 4. How does row or column swapping affect the determinant? 5. Can the determinant be computed for larger matrices using a similar method? **Tip**: Determinants are useful for solving linear systems of equations via Cramer's Rule.

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Math Problem Analysis

Mathematical Concepts

Determinants
Matrix Algebra

Formulas

Determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Theorems

Determinant Expansion (Laplace Expansion)

Suitable Grade Level

Grades 10-12