Math Problem Statement
Find the inverse of the given matrix A and verify the product A^-1 * A equals the identity matrix.
Solution
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The matrix is:
1 & 3 & 2 \\ -3 & -3 & -1 \\ 2 & 1 & 0 \end{bmatrix}$$ ### Steps: 1. **Check if the matrix is invertible** by finding its determinant ($$\text{det}(A)$$). 2. **Compute the inverse $$A^{-1}$$** using the formula $$A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)$$, where $$\text{adj}(A)$$ is the adjugate of $$A$$. 3. **Verify the result** by calculating \( A \ # Define the matrix A A = np.array([[1, 3, 2], [-3, -3, -1], [2, 1, 0]]) # Calculate the determinant of A det_A = np.linalg.det(A) # Check if the matrix is invertible if det_A != 0: # Calculate the inverse of A A_inv = np.linalg.inv(A) # Verify A_inv * A equals the identity matrix verification = np.dot(A, A_inv) else: A_inv = None ### Results: 1. **Determinant of $$A$$:** $$\text{det}(A) = 1$$ Since the determinant is non-zero, the matrix $$A$$ is invertible. 2. **Inverse of $$A$$ ($$A^{-1}$$):** $$A^{-1} = \begin{bmatrix} 1 & 2 & 3 \\ -2 & -4 & -5 \\ 3 & 5 & 6 \end{bmatrix}$$ 3. **Verification ($$A \times A^{-1}$$):** $$A \times A^{-1} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \text{ (Identity Matrix, approximately)}.$$ Let me know if you want detailed steps for any part of this calculation! --- ### Related Questions: 1. How is the adjugate matrix calculated during inversion? 2. Why is the determinant important for determining invertibility? 3. Can we verify $$A^{-1} \times A = I$$ for consistency? 4. What happens if $$\text{det}(A) = 0$$? 5. How can these operations be extended to larger matrices? **Tip:** Always verify your inverse matrix calculation by multiplying it back withAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Inversion
Linear Algebra
Determinants
Formulas
A^-1 = (1/det(A)) * adj(A)
Verification: A^-1 * A = I (Identity Matrix)
Theorems
Invertible Matrix Theorem
Suitable Grade Level
Undergraduate Level or Advanced High School