Math Problem Statement
Find the inverse of the following matrix by using Gauss-Jordan elimination A = \begin{pmatrix} -2 & 2 & -1 \\ 3 & -5 & 4 \\ 5 & -6 & 4 \end{pmatrix}.
Solution
To solve for the inverse of the given matrix using Gauss-Jordan elimination, follow these steps:
The matrix is:
-2 & 2 & -1 \\ 3 & -5 & 4 \\ 5 & -6 & 4 \end{pmatrix}$$ The inverse of matrix $$A$$, if it exists, can be found by augmenting the matrix $$A$$ with the identity matrix $$I$$: $$[A | I] = \left( \begin{array}{ccc|ccc} -2 & 2 & -1 & 1 & 0 & 0 \\ 3 & -5 & 4 & 0 & 1 & 0 \\ 5 & -6 & 4 & 0 & 0 & 1 \\ \end{array} \right)$$ ### Steps of Gauss-Jordan Elimination: 1. Convert the matrix $$A$$ to reduced row echelon form (RREF). 2. Apply row operations to the augmented matrix until the left side becomes the identity matrix. 3. The right side will become the inverse of $$A$$. I'll perform the Gauss-Jordan elimination process to find the inverse of this matrix.The inverse of the matrix $$A$$ using Gauss-Jordan elimination is: $$A^{-1} = \begin{pmatrix} 4 & -2 & 3 \\ 8 & -3 & 5 \\ 7 & -2 & 4 \end{pmatrix}$$ This is the result of converting the augmented matrix into reduced row echelon form. Would you like a detailed explanation of any of the steps, or do you have any other questions? Here are 5 related questions to expand your understanding: 1. How does Gauss-Jordan elimination differ from Gaussian elimination? 2. What conditions must be satisfied for a matrix to have an inverse? 3. Can all square matrices be inverted using Gauss-Jordan elimination? 4. What are the applications of the inverse matrix in solving systems of equations? 5. How can you check if a matrix is singular before applying Gauss-Jordan elimination? **Tip:** Always ensure the determinant of the matrix is non-zero before attempting to find its inverse. This is a quick way to check if the matrix is invertible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Inversion
Gauss-Jordan Elimination
Formulas
Augmented Matrix [A | I]
Row Reduction to Reduced Row Echelon Form (RREF)
Theorems
Inverse Matrix Theorem
Gauss-Jordan Elimination
Suitable Grade Level
Undergraduate Level
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