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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]| - Mathematics

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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]|`

 
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Solution

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`

Concept: Properties of Determinants
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