Math Problem Statement
If matrix A = [[1, 0, 0], [0, 3, 0], [1, -2, 2]] and matrix B = [[2, 0, 1], [0, -1, -2], [3, 1, -2]], then |AB| = ...
Solution
To solve the problem, we need to calculate the determinant of the product , where and are the given matrices:
Matrix :
1 & 0 & 0 \\ 0 & 3 & 0 \\ 1 & -2 & 2 \end{bmatrix}$$ Matrix $$B$$: $$B = \begin{bmatrix} 2 & 0 & 1 \\ 0 & -1 & -2 \\ 3 & 1 & -2 \end{bmatrix}$$ ### Step 1: Property of Determinants We use the property: $$\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$$ ### Step 2: Calculate $$\text{det}(A)$$ The determinant of $$A$$ is calculated as: $$\text{det}(A) = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 1 & -2 & 2 \end{vmatrix}$$ Expanding along the first row: $$\text{det}(A) = 1 \cdot \begin{vmatrix} 3 & 0 \\ -2 & 2 \end{vmatrix}$$ $$= 1 \cdot [(3 \cdot 2) - (0 \cdot -2)] = 1 \cdot 6 = 6$$ ### Step 3: Calculate $$\text{det}(B)$$ The determinant of $$B$$ is calculated as: $$\text{det}(B) = \begin{vmatrix} 2 & 0 & 1 \\ 0 & -1 & -2 \\ 3 & 1 & -2 \end{vmatrix}$$ Expanding along the first row: $$\text{det}(B) = 2 \cdot \begin{vmatrix} -1 & -2 \\ 1 & -2 \end{vmatrix} - 0 \cdot \begin{vmatrix} 0 & -2 \\ 3 & -2 \end{vmatrix} + 1 \cdot \begin{vmatrix} 0 & -1 \\ 3 & 1 \end{vmatrix}$$ \[ \text{det}(B) = 2 \cdot [(-1 \cdot -2) - (-2 \cdot 1)] + 1 \cdot [(0 \cdot 1) - (-1 \cdot 3)] \] \[ \text{det}(B) = 2 \cdot [2 + 2] + 1 \cdot [0 + 3] = 2 \cdot 4 + 3 = 8 + 3 = 11 \] ### Step 4: Calculate $$\text{det}(AB)$$ Using the property: $$\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$$ $$\text{det}(AB) = 6 \cdot 11 = 66$$ ### Final Answer: The determinant of $$AB$$ is $$\boxed{66}$$. Would you like further details on any of these steps? --- ### Related Questions: 1. What are the steps to calculate determinants for 3x3 matrices? 2. How do determinant properties simplify calculations in linear algebra? 3. Why does $$\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)$$ hold for square matrices? 4. How does matrix multiplication affect the determinant of a matrix product? 5. Can this property be extended to non-square matrices? Why or why not? ### Tip: Always check for row or column simplifications (e.g., zeros) to make determinant calculations faster!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Determinants
Linear Algebra
Formulas
|AB| = |A| * |B|
Determinant formula for a 3x3 matrix
Theorems
Property of determinants for matrix products
Suitable Grade Level
Grades 11-12
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