Find the Inverse of a 3x3 Matrix

Math Problem Statement

Dapatkan invers matrik berikut: [1, 1, 1; 2, -1, 1; 3, -2, -1]

The question asks to find the inverse of the following 3x3 matrix:

1 & 1 & 1 \\ 2 & -1 & 1 \\ 3 & -2 & -1 \end{pmatrix}$$ To find the inverse of a 3x3 matrix, we can use the formula involving the adjugate and the determinant of the matrix. The inverse of a matrix $$A$$ is given by: $$A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)$$ Where: - $$\text{det}(A)$$ is the determinant of the matrix. - $$\text{adj}(A)$$ is the adjugate matrix, which is the transpose of the cofactor matrix. Let me calculate the determinant of the matrix and, if it's non-zero, find its inverse.The determinant of the matrix is 7, which is non-zero. This means the matrix is invertible. Now, I'll compute the inverse of the matrix.The inverse of the matrix is: $$A^{-1} = \begin{pmatrix} 0.4286 & -0.1429 & 0.2857 \\ 0.7143 & -0.5714 & 0.1429 \\ -0.1429 & 0.7143 & -0.4286 \end{pmatrix}$$ Let me know if you'd like further details or explanations! Here are five questions to deepen your understanding: 1. What does it mean for a matrix to be invertible? 2. How do you interpret the determinant in the context of matrix inversion? 3. Can a matrix with a zero determinant have an inverse? Why or why not? 4. How is the cofactor matrix related to finding the inverse of a matrix? 5. Why is it necessary to transpose the cofactor matrix when calculating the adjugate? **Tip:** Always check the determinant first when determining if a matrix has an inverse; a determinant of zero means the matrix is not invertible!

Drop file here or Click Here to upload

Matrix Algebra
Inverse Matrix
Determinants

Inverse of a matrix: A^{-1} = 1/det(A) * adj(A)
Determinant of a 3x3 matrix: det(A)
Adjugate matrix: adj(A)

Matrix Inversion Theorem

Grades 10-12

Find the Inverse of a 3x3 Matrix: [[1, 1, 1], [2, -1, 1], [3, -2, -1]]

Finding the Inverse of a 3x3 Matrix: Step-by-Step Solution

Inverse of 3x3 Matrix A with Determinant and Adjugate Method

How to Calculate the Inverse of a 3x3 Matrix

Find the Inverse of Matrix A = [1 2; -1 3] Using Determinants and Adjugate