Math Problem Statement
Вычислить определитель матрицы по правилу треугольников.
Solution
The problem involves calculating the determinant of a matrix using the "triangle rule." The matrix is:
1 & 1 & 1 \\ 2 & 3 & 4 \\ 4 & 9 & 16 \end{vmatrix}$$ To calculate the determinant using the triangle rule, we proceed by constructing a triangular shape with the matrix elements and calculate the sum of products of diagonals from left to right, subtracting products of diagonals from right to left. ### Step 1: Constructing the diagonals We begin with the matrix and follow the diagonals: #### Diagonals from left to right: 1. $$1 \times 3 \times 16 = 48$$ 2. $$1 \times 4 \times 4 = 16$$ 3. $$1 \times 2 \times 9 = 18$$ The sum of the left-to-right diagonals: $$48 + 16 + 18 = 82$$ #### Diagonals from right to left: 1. $$1 \times 3 \times 4 = 12$$ 2. $$1 \times 4 \times 4 = 16$$ 3. $$1 \times 2 \times 16 = 32$$ The sum of the right-to-left diagonals: $$12 + 16 + 32 = 60$$ ### Step 2: Determinant calculation Now, we subtract the sum of right-to-left diagonals from the sum of left-to-right diagonals: $$\text{Determinant} = 82 - 60 = 22$$ Thus, the determinant of the matrix is **22**. Would you like a more detailed explanation or further help with other matrix-related questions? Here are a few suggestions to explore further: 1. How does the rule of triangles apply to larger matrices? 2. What are some alternative methods for calculating determinants? 3. How do you verify a determinant result? 4. Can the triangle rule be used for non-square matrices? 5. How does the determinant relate to matrix invertibility? **Tip:** The rule of triangles is efficient for 3x3 matrices but becomes more cumbersome for larger matrices, so often methods like cofactor expansion or row reduction are more practical for bigger ones.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Determinants
Linear Algebra
Triangle Rule
Formulas
Determinant formula for 3x3 matrix
Triangle rule for determinants
Theorems
Properties of Determinants
Triangle Rule in 3x3 Determinants
Suitable Grade Level
Grades 10-12