How to Find the Inverse of a 3x3 Matrix

Math Problem Statement

Inverse matrix

The matrix shown in the image appears to be:

1 & -1 & 3 \\ 3 & 0 & 2 \\ 4 & 1 & 0 \end{pmatrix}$$ If you want to find the inverse of this matrix, I'll compute it step by step. ### Step 1: Check if the matrix is invertible We need to calculate the determinant of the matrix. If the determinant is non-zero, the matrix is invertible. Let me calculate the determinant and find the inverse matrix. ### Matrix: $$A = \begin{pmatrix} 1 & -1 & 3 \\ 3 & 0 & 2 \\ 4 & 1 & 0 \end{pmatrix}$$ I'll calculate the determinant and proceed.The determinant of the matrix $$A$$ is approximately $$-1$$, which means the matrix is invertible. Now, I will compute the inverse of the matrix.The inverse of the given matrix is: $$A^{-1} = \begin{pmatrix} 2 & -3 & 2 \\ -8 & 12 & -7 \\ -3 & 5 & -3 \end{pmatrix}$$ Would you like further details or clarifications? Here are 5 related questions you might find interesting: 1. How is the determinant of a 3x3 matrix calculated? 2. What conditions make a matrix non-invertible? 3. Can you find the inverse of any square matrix? 4. How are inverses of matrices used in solving systems of linear equations? 5. What is the geometric interpretation of an inverse matrix? **Tip:** The inverse matrix is useful for solving systems of linear equations, where $$A \mathbf{x} = \mathbf{b}$$, and $$\mathbf{x}$$ can be found as $$\mathbf{x} = A^{-1} \mathbf{b}$$.

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Linear Algebra
Matrix Inversion
Determinants

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

Invertibility Theorem: A matrix is invertible if and only if its determinant is non-zero.

Undergraduate level or advanced high school (Grades 11-12)

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