# Properties of Determinants

1.|At|= |A|

The determinant of matrix A and its transpose At are equal.

2. |A|= 0    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.

r3 = r1 + r2

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

4. If a determinant switches two parallel lines its 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 calculate:

2. Apply the properties of determinants and calculate:

3.Calculate: