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.
r3 = r1 + r2
2. The determinant of matrix A and its transpose At are equal.
|At|= |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:
