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What are the Properties of determinants?


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Solution

Properties of determinants.

There are 10 main properties of determinants which are listed below.

Reflection property:

The determinant remains unchanged if its rows are changed into columns and the columns into rows. This is known as the property of reflection.

All zero property:

If all the elements of a row (or column) are zero, then the determinant is zero.

Proportionality (Repetition) Property:

If all elements of a row (or column) are proportional (identical) to the elements of some other row (or column), then the determinant is zero.

Switching Property:

The interchange of any two rows (or columns) of the determinant changes its sign.

Scalar Multiple Property:

If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.

Sum property:

a+mbcd+nefg+ohi=abcdefghi+mbcnefohi

Property of Invariance:

a1b1c1a2b2c2a3b3c3=a1+αb1+βc1b1c1a2+αb2+βc2b2c2a3+αb3+βc3b3c3

A determinant remains unchanged under an operation of the form Ci→Ci+αCj+βCk , where j,k≠i

or an operation has form Ri→Ri+αRj+βRk , where j,k≠i.

Factor Property:

If a determinant becomes zero when we put x=α. Then, (x-α) is a factor of the determinant.

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.

a1b1c10b2c200c3=a100a2b20a3b3c3=a1b2c3

Determinant of cofactor matrix:

∆=a11a12a13a21a22a23a31a32a33 then ∆1=C11C12C13C21C22C23C31C32C33 , where Cij denotes the cofactor of the element aij in ∆.


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