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## Solution example

**First Line Decomposition**

To calculate the determinant of a matrix by first line decomposition, it is necessary to multiply each element of the given line by the corresponding minor;

The minors corresponding to a certain element are found by eliminating i-row , j-column from the matrix A, after which we find the determinant of the resulting matrix;

i, j are the numbers of the row and the column, in which the certain element is located;

After calculating the products of each element of the first row, in the corresponding minor, it is necessary to add and subtract them;

The sign of addition and subtraction changes in order, starting with the sign of addition;

Near the first product there is a plus sign, near the second a minus sign, etc.

000 | 71 | 8 | 8 | 2 | 000 |

7 | 8 | 5 | 2 | ||

2 | 5 | 8 | 7 | ||

4 | 5 | 5 | 2 |

_{11}* A

_{11}- a

_{12}* A

_{12}+ a

_{13}* A

_{13}- a

_{14}* A

_{14};

So we find the minors of each element of the first row.

Find the minor element with index 11

To do this, it is necessary to exclude the 1 row and the 1 column from matrix A, after which we obtain the following matrix:

000 | 8 | 5 | 2 | 000 |

5 | 8 | 7 | ||

5 | 5 | 2 |

Next, we calculate the determinant of this matrix.

It is equal to -57, this is the minor of element 11.

_{11}=

000 | 71 | 8 | 8 | 2 | 000 |

7 | 8 | 5 | 2 | ||

2 | 5 | 8 | 7 | ||

4 | 5 | 5 | 2 |

000 | 8 | 5 | 2 | 000 |

5 | 8 | 7 | ||

5 | 5 | 2 |

Find the minor element with index 12

To do this, it is necessary to exclude the 1 row and the 2 column from matrix A, after which we obtain the following matrix:

000 | 7 | 5 | 2 | 000 |

2 | 8 | 7 | ||

4 | 5 | 2 |

Next, we calculate the determinant of this matrix.

It is equal to -57, this is the minor of element 12.

_{12}=

000 | 71 | 8 | 8 | 2 | 000 |

7 | 8 | 5 | 2 | ||

2 | 5 | 8 | 7 | ||

4 | 5 | 5 | 2 |

000 | 7 | 5 | 2 | 000 |

2 | 8 | 7 | ||

4 | 5 | 2 |

Find the minor element with index 13

To do this, it is necessary to exclude the 1 row and the 3 column from matrix A, after which we obtain the following matrix:

000 | 7 | 8 | 2 | 000 |

2 | 5 | 7 | ||

4 | 5 | 2 |

Next, we calculate the determinant of this matrix.

It is equal to -3, this is the minor of element 13.

_{13}=

000 | 71 | 8 | 8 | 2 | 000 |

7 | 8 | 5 | 2 | ||

2 | 5 | 8 | 7 | ||

4 | 5 | 5 | 2 |

000 | 7 | 8 | 2 | 000 |

2 | 5 | 7 | ||

4 | 5 | 2 |

Find the minor element with index 14

To do this, it is necessary to exclude the 1 row and the 4 column from matrix A, after which we obtain the following matrix:

000 | 7 | 8 | 5 | 000 |

2 | 5 | 8 | ||

4 | 5 | 5 |

Next, we calculate the determinant of this matrix.

It is equal to 21, this is the minor of element 14.

_{14}=

000 | 71 | 8 | 8 | 2 | 000 |

7 | 8 | 5 | 2 | ||

2 | 5 | 8 | 7 | ||

4 | 5 | 5 | 2 |

000 | 7 | 8 | 5 | 000 |

2 | 5 | 8 | ||

4 | 5 | 5 |

Now we need to calculate the product of the first element by its corresponding minor.

71 * (-57) = -4047;

Further, from this product, it is necessary to subtract the product of the second element by the corresponding minor.

-4047 - (8 * (-57)) = -4047 - (-456) = -3591;

Now to the result you need to add the product of the third element to the corresponding minor.

-3591 (8 * (-3)) = -3591 (-24) = -3615;

And finally, from the obtained result , it is necessary to subtract the product of the fourth element by the corresponding minor

-3615 - (2 * 21) = -3615 - 42 = -3657;

The result of this subtraction is the determinant of the matrix A

**Answer:**det(A) = -3657

**Sarrus**

000 | 2 | 5 | 6 | 000 |

5 | 8 | 2 | ||

3 | 5 | 7 |

000 | 2 | 5 | 6 | 000 | 2 | 5 |

5 | 8 | 2 | 5 | 8 | ||

3 | 5 | 7 | 3 | 5 |

_{11}a

_{22}a

_{33}) + (a

_{12}a

_{23}a

_{31}) + (a

_{13}a

_{21}a

_{32}) -

_{13}a

_{22}a

_{31}) - (a

_{11}a

_{23}a

_{32}) - (a

_{12}a

_{21}a

_{33}) =

**Answer:**det(A) = -47

**Reduction to Triangular Form**

000 | 71 | 8 | 8 | 2 | 000 |

7 | 8 | 5 | 2 | ||

2 | 5 | 8 | 7 | ||

4 | 5 | 5 | 2 |

000 | 71 | 8 | 8 | 2 | 000 |

0 | 7.21128 | 4.21128 | 1.80282 | ||

0 | 4.77464 | 7.77464 | 6.94366 | ||

0 | 4.54928 | 4.54928 | 1.88732 |

000 | 71 | 8 | 8 | 2 | 000 |

0 | 7.21128 | 4.21128 | 1.80282 | ||

0 | 0 | 4.98631 | 5.74999 | ||

0 | 0 | 1.89255 | 0.74999 |

000 | 71 | 8 | 8 | 2 | 000 |

0 | 7.21128 | 4.21128 | 1.80282 | ||

0 | 0 | 4.98631 | 5.74999 | ||

0 | 0 | 0 | -1.43242 |

**Answer:**det(A) = -3657