Properties of Determinants

1. A determinant is zero if:

It has two equal lines

Properties of Determinants

All elements of a line are zero.

Properties of Determinants

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

Properties of Determinants

r3 = r1 + r2


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

|At|= |A|

Transpose of a Matrix

Transpose of a Matrix

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

Triangular Determinant


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

Determinant


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.

Determinant Determinant


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

Determinant Multiplication


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.

Properties of 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:

Properties of Determinants          Properties of Determinants        Properties of Determinants


Properties of Determinants

Properties of Determinants

Properties of Determinants


2. Apply the properties of determinants and solve:

Properties of Determinants

Properties of Determinants

Properties of Determinants

Properties of Determinants


3.Calculate:

Properties of Determinants      Properties of Determinants

Properties of Determinants

Properties of Determinants





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