# Properties of Determinants

1. A determinant is zero if:

It has two equal lines

All elements of a line are zero.

The elements of a line are a linear combination of the others.

**r _{3} = r_{1} + r_{2}**

2. The determinant of matrix A and its transpose A^{t} are equal.

**|A ^{t}|= |A| **

3.A triangular determinant is the product of the diagonal elements.

4. If a determinant switches two parallel lines, the determinant changes sign.

5. If the elements of a line are added to the elements of another parallel line previously multiplied by a real number, the value of the determinant is unchanged.

6.If a determinant is multiplied by a real number, any line can be multiplied by the above mentioned number, but only one.

7. If all the elements of a line or column are formed by two addends, the above mentioned determinant decomposes in the sum of two determinants.

8.
**|A·B| =|A|·|B| **

The determinant of a product equals the product of the determinants.

#### Examples

1. Apply the properties of determinants and solve:

2. Apply the properties of determinants and solve:

3.Calculate: