Math Problem Statement
Given the matrix A = [[1, 1, 2], [0, 1, -1], [-2, -2, -4]], what is the determinant of A?
Solution
To calculate the determinant of the given matrix :
1 & 1 & 2 \\ 0 & 1 & -1 \\ -2 & -2 & -4 \end{bmatrix}$$ We use the determinant formula for a 3x3 matrix: $$\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)$$ For matrix \( A = \begin{bmatrix} 1 & 1 & 2 \\ 0 & 1 & -1 \\ -2 & -2 & -4 \end{bmatrix} \): - $$a = 1, b = 1, c = 2$$ - First row $$1, 1, 2$$, second row $$0, 1, -1$$, third row $$-2, -2, -4$$. Expanding the determinant: \[ \text{det}(A) = 1 \cdot [(1)(-4) - (-1)(-2)] - 1 \cdot [(0)(-4) - (-1)(-2)] + 2 \cdot [(0)(-2) - (1)(-2)] \] Simplify each term: 1. $$(1)(-4) - (-1)(-2) = -4 - 2 = -6$$ 2. $$(0)(-4) - (-1)(-2) = 0 - 2 = -2$$ 3. $$(0)(-2) - (1)(-2) = 0 + 2 = 2$$ Now substitute back: $$\text{det}(A) = 1(-6) - 1(-2) + 2(2)$$ $$\text{det}(A) = -6 + 2 + 4$$ $$\text{det}(A) = 0$$ The determinant of the matrix $$A$$ is $$\boxed{0}$$. Let me know if you'd like further clarifications or have any questions! Here are related questions to consider: 1. How does a zero determinant affect the invertibility of a matrix? 2. What does a determinant of zero indicate geometrically about a matrix? 3. Can we compute the determinant of a larger matrix (4x4 or higher) efficiently? 4. What role does row reduction play in determinant calculation? 5. How does the determinant change if one row of a matrix is multiplied by a scalar? **Tip:** For a quick check of determinant, observe if rows or columns are linearly dependent—this often makes the determinant zero.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Determinants
Formulas
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
A matrix is singular if and only if its determinant is zero
Properties of determinants (linear dependence of rows/columns)
Suitable Grade Level
Grades 11-12, Undergraduate