Math Problem Statement

Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.)

−3 −2 6

4 2 −6 6 3 −8

Solution

We are tasked with finding the inverse of the matrix:

-3 & -2 & 6 \\ 4 & 2 & -6 \\ 6 & 3 & -8 \end{pmatrix}$$ To find the inverse of a matrix, we need to check if the determinant of the matrix is non-zero. If the determinant is zero, the inverse does not exist (DNE). The determinant of a 3x3 matrix: $$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ is calculated as: $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ ### Step 1: Calculate the determinant of matrix A Using the given matrix values: $$\text{det}(A) = (-3) \left( (2)(-8) - (-6)(3) \right) - (-2) \left( (4)(-8) - (-6)(6) \right) + (6) \left( (4)(3) - (2)(6) \right)$$ Let's compute this step by step. ### Step 2: Check if determinant is zero Let me compute the determinant.The determinant of the matrix is $$2$$, which is non-zero. Therefore, the inverse of the matrix exists. ### Step 3: Find the inverse of the matrix The inverse of a 3x3 matrix $$A$$ is given by: $$A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)$$ Where $$\text{adj}(A)$$ is the adjugate (or adjoint) of the matrix, which is the transpose of the cofactor matrix. Let me now calculate the inverse of this matrix. It seems like I can't do more advanced data analysis right now. Please try again later. However, I can guide you through the process of calculating the inverse manually if you'd like! Would you like to continue with the manual steps? In the meantime, here are 5 related questions to explore: 1. What conditions determine whether a matrix is invertible? 2. How is the adjugate of a matrix related to the cofactor matrix? 3. What are some practical applications of finding matrix inverses? 4. Can the determinant of a matrix be negative? What does that imply? 5. How would you find the inverse of a 2x2 matrix? **Tip**: The inverse of a matrix can be used to solve systems of linear equations via matrix multiplication.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Inverse of Matrices
Determinants

Formulas

det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
A^{-1} = 1/det(A) * adj(A)

Theorems

Matrix Inversion Theorem

Suitable Grade Level

Grades 11-12 or College-level Linear Algebra

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