Inversa de una matriz de 3x3 método de Gauss Jordan | Ejemplo 2

Summary

The video demonstrates how to find the inverse of a (3\times3) matrix using Gauss–Jordan elimination. The key idea is to transform the original matrix into the identity matrix. The same row operations transform the identity matrix into the original matrix’s inverse.

Method

1. Form an augmented matrix: Write the original matrix (A) on the left and the (3\times3) identity matrix (I) on the right: [ [A\mid I]. ]

2. Use row operations to turn the left side into the identity matrix:

   • Eliminate entries below the first pivot.
   • Eliminate entries below the next pivot.
   • Work upward to eliminate entries above the pivots.
   • Scale each row so the main diagonal entries are (1).

3. Apply every operation to the entire row, including the entries on the right side of the augmented matrix. Useful operations include multiplying a row by a number and adding a multiple of one row to another.

4. Choose multipliers that cancel the target entry. The instructor illustrates finding equal-and-opposite values so their sum is zero. When whole-number multipliers are inconvenient, multiplying by the other entry’s value can create matching terms that cancel.

5. Read the result: If the left side becomes (I), the right side is (A^{-1}). If the left side cannot be reduced to (I), this process does not produce an inverse.

In the worked example, the operations imply that the starting matrix is [ \begin{pmatrix} 2&1&3\ -1&2&4\ 0&1&3 \end{pmatrix}, ] and its inverse is given as [ \begin{pmatrix} \frac12&0&-\frac12\ \frac34&\frac32&-\frac{11}{4}\ 0&-\frac14&\frac54 \end{pmatrix}. ]

Some arithmetic in the auto-generated subtitles is garbled, but the method and row-operation sequence are clear.

Checking and Practice

• Verify the result by multiplying the original matrix by the proposed inverse. The product should be the identity matrix.
• The video ends with another (3\times3) practice problem. The instructor outlines successive row operations and notes that different valid choices of row operations can lead to the same inverse.

Speakers or Sources Featured

• Presenter/instructor: The mathematics teacher from the Matemáticas Profe Alex channel.
• Other speakers: None identified.