PROPERTIES OF DETERMINANTS

Property I

propertiesofdeterminants1

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





For example: 
propertiesofdeterminants2
                         = 1 (0 - 4) – 2 (0 – (-1)) + 3 (8 - 0)
                         = -4 -2 + 24 = 18 … (i)
Let propertiesofdeterminants3
         = 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
propertiesofdeterminants4
     = 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: propertiesofdeterminants5
                       
                        = 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
propertiesofdeterminants6

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.  

propertiesofdeterminants7

Property VI

The value of the determinant remains same if we apply the operation propertiesofdeterminants8
For example: Using properties of determinants: Solve  


propertiesofdeterminants9

  1. propertiesofdeterminants10
                            = (ab + ac + bc) x 0
                            = 0
                                            

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