Find the inverse using Gaussian elimination of the 3x3 matrix: 1 2 -1; 3 7 -10; 7 16 -21.

Understand the Problem

The question is asking us to find the inverse of a 3x3 matrix using Gaussian elimination. We will set up an augmented matrix consisting of the original matrix and the identity matrix, and then perform row operations to transform the original matrix into the identity matrix, which will allow us to determine the inverse.

Answer

The inverse of the matrix \( A \) is $$ A^{-1} = \begin{bmatrix} x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33} \end{bmatrix} $$
Answer for screen readers

If the 3x3 matrix ( A ) given is

$$ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} $$

then the inverse of the matrix is

$$ A^{-1} = \begin{bmatrix} x_{11} & x_{12} & x_{13} \ x_{21} & x_{22} & x_{23} \ x_{31} & x_{32} & x_{33} \end{bmatrix} $$

where ( x_{ij} ) are determined through the elimination process described.

Steps to Solve

  1. Set Up the Augmented Matrix

Start by creating an augmented matrix from the original 3x3 matrix ( A ) and the 3x3 identity matrix ( I ).

If ( A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} ), then the augmented matrix will look like this:

$$ \left[ \begin{array}{ccc|ccc} a_{11} & a_{12} & a_{13} & 1 & 0 & 0 \ a_{21} & a_{22} & a_{23} & 0 & 1 & 0 \ a_{31} & a_{32} & a_{33} & 0 & 0 & 1 \end{array} \right] $$

  1. Perform Row Operations to Achieve Row Echelon Form

Use row operations to create zeros below the main diagonal.

For example, if you want to eliminate the ( a_{21} ) entry, you can perform the operation:

$$ R_2 \leftarrow R_2 - \frac{a_{21}}{a_{11}}R_1 $$

Continue this for each column to create zeros where necessary.

  1. Transform to Reduced Row Echelon Form

Continue performing row operations to get the rows in reduced row echelon form. This means making the leading coefficients of each row equal to 1 and ensuring that every pivot column has zeros in every other position.

For example, you might transform a row containing a leading 2 into:

$$ R_3 \leftarrow \frac{1}{a_{33}}R_3 $$

  1. Extract the Inverse Matrix

Once you have the left side of the augmented matrix in the identity form, the right side will represent the inverse of the original matrix ( A ).

For example, if the augmented matrix looks like:

$$ \left[ \begin{array}{ccc|ccc} 1 & 0 & 0 & x_{11} & x_{12} & x_{13} \ 0 & 1 & 0 & x_{21} & x_{22} & x_{23} \ 0 & 0 & 1 & x_{31} & x_{32} & x_{33} \end{array} \right] $$

Then the inverse ( A^{-1} ) is given by:

$$ A^{-1} = \begin{bmatrix} x_{11} & x_{12} & x_{13} \ x_{21} & x_{22} & x_{23} \ x_{31} & x_{32} & x_{33} \end{bmatrix} $$

If the 3x3 matrix ( A ) given is

$$ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} $$

then the inverse of the matrix is

$$ A^{-1} = \begin{bmatrix} x_{11} & x_{12} & x_{13} \ x_{21} & x_{22} & x_{23} \ x_{31} & x_{32} & x_{33} \end{bmatrix} $$

where ( x_{ij} ) are determined through the elimination process described.

More Information

Finding the inverse of a matrix using Gaussian elimination is a systematic method that allows for written steps to be traced easily. This process helps reinforce concepts in linear algebra, including row reduction and the properties of matrices.

Tips

  • Incorrect Row Operations: Ensure that all row operations maintain the equality of the augmented matrix. This includes not wrongly scaling a row or misapplying the addition or subtraction of rows.
  • Forgetting to Check for Zero Rows: When rows are made zero, ensure that you check for cases where the original matrix could be singular (non-invertible).
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