# Property I

The value of the determinant remains unchanged if its rows and columns are interchanged.

For example:

= 1 (0 - 4) – 2 (0 – (-1)) + 3 (8 - 0)
= -4 -2 + 24 = 18 … (i)
Let
= 1 (0 - 4) - 2 (0 - 12) – 1 (2 - 0)
= -4 + 24 – 2 = 18
Hence, ∆ = ∆’

## Property II

If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.
For example:  We know ∆ = 18 from [equation (i) above]
Let us interchange 2nd and 3rd rows of ∆’ and find its value

= 1 (4 - 0) - 2 (12 - 0) – 1 (0 - 2)
= 4 – 24 + 2 = -18
Hence, ∆’ = -∆

## Property III

If any two rows (or columns) of a determinant are identical, then the value of the determinant is zero.
For example:

= 2[-1 - 0] - 0[-1 - 8] - 1[0 - 2]
= -2 – 0 + 2 = 0

## Property IV

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 V

If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

## Property VI

The value of the determinant remains same if we apply the operation
For example: Using properties of determinants: Solve

1.                         = (ab + ac + bc) x 0
= 0

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