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