** Example 1**.

For a matrix

find .

** Solution**:

In the order, we apply the following transformations:

(1.) interchange the first and the second row,

(2.) the first row multiplied by added to second row.

(3.) the second row multiplied by added to first row,

(4.) the first row multiplied by .

An inverse matrix of the given matrix is

** Example 2**.

Using the elementary transformations of matrices, find the inverse matrix of matrix

* Solution*:

With we will denote a row in which we perform elementary transformations.

Therefore, an inverse matrix of a matrix is

** Example 3**.

Solve the following system of equations by using the Cramer’s rule:

** Solution**:

Firstly, we write the system in a matrix form:

Let a matrix be a matrix of the system above, that is

The determinant of a matrix is (check it!).

Now we need to calculate the determinant of a matrix in which the first column is replaced with the column matrix of free coefficients, that is

The determinant of a matrix is .

Analogously we treat for the remaining columns of the matrix – determinants of matrices that we get, in the order amounts , , . According to the Cramer’s rule, we get the final solutions:

** Example 4**.

Using the elementary transformations, solve the following system of equations:

** Solution**:

The rank of a matrix (matrix of the system) and augmented matrix is . Therefore, the solution depends on the two free parameters. We have (from the system above):

If and ; , then

The solution we can write in a matrix form:

*Example 5.*

Solve the following system of equations:

** Solution**:

The -th row gives the equation , which is not possible, therefore, the given system of equations does not have solutions.

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