Math Problem Statement

find inverse of the given matix a in gauss jordan method

Find the inverse of the given matrix A using the Gauss-Jordan method.

Solution

The given matrix AA is:

A=[03−4567208]A = \begin{bmatrix} 0 & 3 & -4 \\ 5 & 6 & 7 \\ 2 & 0 & 8 \end{bmatrix}

To find the inverse of the matrix AA using the Gauss-Jordan method, we will perform row operations to transform AA into the identity matrix, while simultaneously applying these operations to the identity matrix to obtain A−1A^{-1}.

Steps to Find the Inverse Using the Gauss-Jordan Method:

  1. Augment the matrix AA with the identity matrix:
0 & 3 & -4 & 1 & 0 & 0 \\ 5 & 6 & 7 & 0 & 1 & 0 \\ 2 & 0 & 8 & 0 & 0 & 1 \end{array}\right]$$ 2. **Use row operations to make the augmented matrix into reduced row echelon form**: - First, swap Row 1 and Row 2 to bring a non-zero element to the top left: $$\left[\begin{array}{ccc|ccc} 5 & 6 & 7 & 0 & 1 & 0 \\ 0 & 3 & -4 & 1 & 0 & 0 \\ 2 & 0 & 8 & 0 & 0 & 1 \end{array}\right]$$ - Normalize Row 1 by dividing by 5: $$\left[\begin{array}{ccc|ccc} 1 & 1.2 & 1.4 & 0 & \frac{1}{5} & 0 \\ 0 & 3 & -4 & 1 & 0 & 0 \\ 2 & 0 & 8 & 0 & 0 & 1 \end{array}\right]$$ - Eliminate the first column below Row 1: - For Row 3: $$R3 = R3 - 2 \times R1$$ $$\left[\begin{array}{ccc|ccc} 1 & 1.2 & 1.4 & 0 & \frac{1}{5} & 0 \\ 0 & 3 & -4 & 1 & 0 & 0 \\ 0 & -2.4 & 5.2 & 0 & -\frac{2}{5} & 1 \end{array}\right]$$ - Next, normalize Row 2 by dividing by 3: $$\left[\begin{array}{ccc|ccc} 1 & 1.2 & 1.4 & 0 & \frac{1}{5} & 0 \\ 0 & 1 & -\frac{4}{3} & \frac{1}{3} & 0 & 0 \\ 0 & -2.4 & 5.2 & 0 & -\frac{2}{5} & 1 \end{array}\right]$$ - Eliminate Row 2's contribution from Row 1 and Row 3: - For Row 1: $$R1 = R1 - 1.2 \times R2$$ - For Row 3: $$R3 = R3 + 2.4 \times R2$$ After the above operations: $$\left[\begin{array}{ccc|ccc} 1 & 0 & 3 & -0.4 & \frac{1}{5} & 0 \\ 0 & 1 & -\frac{4}{3} & \frac{1}{3} & 0 & 0 \\ 0 & 0 & 2 & \frac{4}{5} & -\frac{2}{5} & 1 \end{array}\right]$$ - Finally, normalize the last row and adjust the first two rows: - Normalize Row 3 by dividing by 2: $$\left[\begin{array}{ccc|ccc} 1 & 0 & 3 & -0.4 & \frac{1}{5} & 0 \\ 0 & 1 & -\frac{4}{3} & \frac{1}{3} & 0 & 0 \\ 0 & 0 & 1 & \frac{2}{5} & -\frac{1}{5} & \frac{1}{2} \end{array}\right]$$ - Adjust Rows 1 and 2 to eliminate the third column: - For Row 1: $$R1 = R1 - 3 \times R3$$ - For Row 2: $$R2 = R2 + \frac{4}{3} \times R3$$ $$\left[\begin{array}{ccc|ccc} 1 & 0 & 0 & -2 & 1 & -\frac{3}{2} \\ 0 & 1 & 0 & 1 & -\frac{1}{2} & 1 \\ 0 & 0 & 1 & \frac{2}{5} & -\frac{1}{5} & \frac{1}{2} \end{array}\right]$$ Thus, the inverse of matrix $$A$$ is: $$A^{-1} = \begin{bmatrix} -2 & 1 & -\frac{3}{2} \\ 1 & -\frac{1}{2} & 1 \\ \frac{2}{5} & -\frac{1}{5} & \frac{1}{2} \end{bmatrix}$$ Do you have any questions or need further details? ### Additional Related Questions: 1. What are the conditions for a matrix to have an inverse? 2. How can you verify that a matrix inverse is correct? 3. What are other methods to find the inverse of a matrix apart from the Gauss-Jordan method? 4. What does it mean if a matrix does not have an inverse? 5. How does the determinant of a matrix relate to the existence of its inverse? **Tip:** Always check the determinant of the matrix first; if it's zero, the matrix does not have an inverse.

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Math Problem Analysis

Mathematical Concepts

Matrix Inversion
Linear Algebra
Gauss-Jordan Elimination

Formulas

Inverse of matrix A: A^{-1}
Gauss-Jordan elimination steps
Row operations for matrix transformations

Theorems

Gauss-Jordan Theorem
Determinant for invertibility

Suitable Grade Level

Grades 11-12, College

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