Question

 

Using properties of determinants, prove that 

`|[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]|=2|[a,b,c],[p,q,r],[x,y,z]|`

 

Answer 1

Operating C₁→ C₁-(C+ C₃ ), we get

LHS= `|[-2a,c+a,a+b],[-2p,r+p,q+p],[-2x,z+z,x+y]|`

`=-2|[a,c+a,a+b],[p,r+p,p+q],[x,z+x,x+y]|`

C₂→C₂-C₁ and C₃→C₃-C₁  ⇒`LHS=-2|[a,c,b],[p,r,q],[x,z,y]|`

C₂↔C₃= `2|[a,b,c],[p,q,r],[x,y,z]|=RHS`