Math Problem Statement
Find the inverse of the matrix B =
[[-2, 4, 2, -1],
[2, 3, 4, 3],
[3, -1, 3, 1],
[4, -2, -4, 4]]
Solution
To solve for the inverse of matrix , the matrix provided is:
-2 & 4 & 2 & -1 \\ 2 & 3 & 4 & 3 \\ 3 & -1 & 3 & 1 \\ 4 & -2 & -4 & 4 \end{pmatrix}$$ We can calculate the inverse of a matrix by applying the formula: $$B^{-1} = \frac{1}{\det(B)} \cdot \text{adj}(B)$$ Where: - $$\det(B)$$ is the determinant of matrix $$B$$. - $$\text{adj}(B)$$ is the adjugate of matrix $$B$$, which is the transpose of its cofactor matrix. Let me calculate the inverse for you step-by-step. I'll first compute the determinant of matrix $$B$$ to check if it's invertible (i.e., if $$\det(B) \neq 0$$). Then, I'll find the adjugate and the inverse.The determinant of matrix $$B$$ is approximately $$132$$. Since the determinant is non-zero, the matrix is invertible. Next, I will compute the adjugate of matrix $$B$$ and then find the inverse.The inverse of matrix $$B$$ is: $$B^{-1} = \begin{pmatrix} 0.652 & -0.439 & 0.606 & 0.341 \\ 0.545 & -0.182 & 0.182 & 0.227 \\ -0.258 & 0.197 & -0.030 & -0.205 \\ -0.636 & 0.545 & -0.545 & -0.182 \end{pmatrix}$$ Would you like further explanation on how the adjugate or determinant was calculated, or do you have any other questions? Here are 5 related questions to expand your understanding: 1. What is the determinant of a 3x3 matrix, and how is it computed? 2. How do you find the adjugate of a matrix in detail? 3. What are the conditions under which a matrix is not invertible? 4. Can you explain the difference between the cofactor matrix and the adjugate matrix? 5. How can matrix inversion be applied in solving systems of linear equations? **Tip:** When working with large matrices, using numerical tools like Python or calculators can speed up the inversion process.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Formulas
Inverse of a matrix formula: B^{-1} = 1/det(B) * adj(B)
Determinant of a matrix
Adjugate of a matrix
Theorems
A matrix is invertible if and only if its determinant is non-zero.
The adjugate matrix is the transpose of the cofactor matrix.
Suitable Grade Level
Undergraduate Level (Linear Algebra)