Give all the property of Determinants .with examples
Don't give wrong and meaningless answer.
Answers
Properties of Determinants
Property 1
The value of the determinant remains unchanged if both rows and columns are interchanged.
Property 2:
If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.
Property 3:
If any two rows (or columns) of a determinant are identical (all corresponding elements are same), then the value of the determinant is zero.
Property 4:
If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.
Property 5:
If some or all elements of a row or column of a determinant are expressed as the sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants.
Property 6:
If the equimultiples of corresponding elements of other rows (or columns) are added to every element of any row or column of a determinant, then the value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation Ri → Ri + k Rj or Ci → Ci + k Cj .
Answer:
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Step-by-step explanation:
1. Reflection Property:
The determinant remains unaltered if its rows are changed into columns and the columns into rows. This is known as the property of reflection.
2. All-zero Property:
If all the elements of a row (or column) are zero, then the determinant is zero.
3. Proportionality (Repetition) Property:
If the all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.
4. Switching Property:
The interchange of any two rows (or columns) of the determinant changes its sign.
5 Factor Property:
If a determinant Δ becomes zero when we put x=α, then (x−α) is a factor of Δ.
6 Triangle Property:
If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements.
