Basic Properties of Determinants

2.  Using Properties of Determinant, Prove That

∣∣∣∣adgbehcfi∣∣∣∣|abcdefghi|

=

∣∣∣∣bachgiedf∣∣∣∣|bheagdcif|

Solution: Interchanging the rows and columns across the diagonals by making use of reflection property and then using the switching property of determination we can get the desired outcome.

L.H.S =

∣∣∣∣adgbehcfi∣∣∣∣|abcdefghi|

=

∣∣∣∣abcdefghi∣∣∣∣|adgbehcfi|

(Interchanging rows and columns across the diagonals)

= (-1)

∣∣∣∣abcghidef∣∣∣∣|agdbhecif|

= (1)² =

∣∣∣∣bachgiedf∣∣∣∣|bheagdcif|

=

∣∣∣∣bachgiedf∣∣∣∣|bheagdcif|

= R.H.S

Quiz Time

1. According to the Determinant Properties, the Value of Determinant Equals to Zero if Row is

1. 
   Multiplied by row
   
2. 
   Multiplied to column
   
3. 
   Divided to row
   
4. 
   Divided to column