Math Problem Statement
Solve
If A = [[1, 1, -2], [2, 1, -3], [5, 4, -9]], find |A|.
Solution
We need to calculate the determinant of the given matrix :
The determinant of a matrix is calculated using the formula: where the matrix is:
a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ For $$A$$: $$a = 1, \, b = 1, \, c = -2, \, d = 2, \, e = 1, \, f = -3, \, g = 5, \, h = 4, \, i = -9$$ Substitute these values into the formula: \[ |A| = 1[(1)(-9) - (-3)(4)] - 1[(2)(-9) - (-3)(5)] + (-2)[(2)(4) - (1)(5)] \] Simplify each term step-by-step: 1. For $$1[(1)(-9) - (-3)(4)]$$: $$(1)(-9) - (-3)(4) = -9 + 12 = 3$$ So, $$1 \cdot 3 = 3$$. 2. For $$-1[(2)(-9) - (-3)(5)]$$: $$(2)(-9) - (-3)(5) = -18 + 15 = -3$$ So, $$-1 \cdot (-3) = 3$$. 3. For $$(-2)[(2)(4) - (1)(5)]$$: $$(2)(4) - (1)(5) = 8 - 5 = 3$$ So, $$(-2) \cdot 3 = -6$$. Now sum all these results: $$|A| = 3 + 3 - 6 = 0$$ Thus, the determinant of the matrix is: $$|A| = 0$$ --- Do you have any questions or would you like a detailed explanation of any step? --- ### Related Questions 1. What does it mean if the determinant of a matrix is zero? 2. How can you verify the determinant using row reduction? 3. What are the properties of determinants for triangular matrices? 4. How do cofactors help in determinant calculations? 5. Can a matrix with a zero determinant have an inverse? ### Tip Always check for row or column patterns (e.g., proportionality) to simplify determinant calculations!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Determinants
Formulas
|A| = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Determinant expansion for 3x3 matrices
Suitable Grade Level
Grades 11-12 or Undergraduate