Math Problem Statement

find the inverse of the following matrix using row operation method 2 -3 3 A= 2 2 3 3 -2 2

Solution

To find the inverse of the given matrix AA using row operations, follow these steps:

Given matrix:

A=[233223322]A = \begin{bmatrix} 2 & -3 & 3 \\ 2 & 2 & 3 \\ 3 & -2 & 2 \end{bmatrix}

We augment AA with the identity matrix II:

[AI]=[233100223010322001][A | I] = \begin{bmatrix} 2 & -3 & 3 & | 1 & 0 & 0 \\ 2 & 2 & 3 & | 0 & 1 & 0 \\ 3 & -2 & 2 & | 0 & 0 & 1 \end{bmatrix}

Now, perform row operations to convert the left side into the identity matrix.

Step 1: Make the first pivot 1

Divide row 1 by 2:

[11.51.50.500223010322001]\begin{bmatrix} 1 & -1.5 & 1.5 & | 0.5 & 0 & 0 \\ 2 & 2 & 3 & | 0 & 1 & 0 \\ 3 & -2 & 2 & | 0 & 0 & 1 \end{bmatrix}

Step 2: Make other elements in column 1 zero

Subtract 2×R12 \times R_1 from R2R_2 and 3×R13 \times R_1 from R3R_3:

[11.51.50.50005011002.52.51.501]\begin{bmatrix} 1 & -1.5 & 1.5 & | 0.5 & 0 & 0 \\ 0 & 5 & 0 & | -1 & 1 & 0 \\ 0 & 2.5 & -2.5 & | -1.5 & 0 & 1 \end{bmatrix}

Step 3: Make the second pivot 1

Divide row 2 by 5:

[11.51.50.5000100.20.2002.52.51.501]\begin{bmatrix} 1 & -1.5 & 1.5 & | 0.5 & 0 & 0 \\ 0 & 1 & 0 & | -0.2 & 0.2 & 0 \\ 0 & 2.5 & -2.5 & | -1.5 & 0 & 1 \end{bmatrix}

Step 4: Make other elements in column 2 zero

Subtract 2.5×R22.5 \times R_2 from R3R_3 and add 1.5×R21.5 \times R_2 to R1R_1:

[101.50.20.300100.20.20002.510.51]\begin{bmatrix} 1 & 0 & 1.5 & | 0.2 & 0.3 & 0 \\ 0 & 1 & 0 & | -0.2 & 0.2 & 0 \\ 0 & 0 & -2.5 & | -1 & -0.5 & 1 \end{bmatrix}

Step 5: Make the third pivot 1

Divide row 3 by -2.5:

[101.50.20.300100.20.200010.40.20.4]\begin{bmatrix} 1 & 0 & 1.5 & | 0.2 & 0.3 & 0 \\ 0 & 1 & 0 & | -0.2 & 0.2 & 0 \\ 0 & 0 & 1 & | 0.4 & 0.2 & -0.4 \end{bmatrix}

Step 6: Make other elements in column 3 zero

Subtract 1.5×R31.5 \times R_3 from R1R_1:

[1000.400.60100.20.200010.40.20.4]\begin{bmatrix} 1 & 0 & 0 & | -0.4 & 0 & 0.6 \\ 0 & 1 & 0 & | -0.2 & 0.2 & 0 \\ 0 & 0 & 1 & | 0.4 & 0.2 & -0.4 \end{bmatrix}

Thus, the inverse of AA is:

A1=[0.400.60.20.200.40.20.4]A^{-1} = \begin{bmatrix} -0.4 & 0 & 0.6 \\ -0.2 & 0.2 & 0 \\ 0.4 & 0.2 & -0.4 \end{bmatrix}


Would you like a verification step to ensure correctness?

Related Questions

  1. How do you check if a matrix is invertible before finding its inverse?
  2. What are alternative methods to find the inverse of a matrix?
  3. Can you find the determinant of AA to confirm its invertibility?
  4. How does the row operation method compare to the adjoint method?
  5. What are some real-world applications of matrix inversion?

Tip:

Always check the determinant before attempting inversion; if det(A)=0\det(A) = 0, the matrix is singular and has no inverse.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Row Operations
Inverse of a Matrix

Formulas

Matrix Inverse using Row Operations
Identity Matrix Augmentation

Theorems

Inverse of a Matrix
Row Reduction Method

Suitable Grade Level

Grades 11-12