" 12."|[b_(1)+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]|

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]|

Prove that the following. [[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]]

Show 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]| .

Prove that abs[[b+c,c+a,a+b],[q+r,r+p,p+q],[y+z,z+x,x+y]]=2abs[[a,b,c],[p,q,r],[x,y,z]]

Without expanding, prove the following |(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)|

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)|

Using the property of determinants and without expanding, prove that: |[b+c, q+r, y+z],[ c+a, r+p, z+x],[ a+b, p+q, x+y]|=2|[a, p, x],[ b, q ,y],[ c, r, z]|

Without expansion, prove the following: |(a+b, b+c, c+a),(p+q, q+r, r+ p),(x+y, y+z, z+x)|=2|(a,b,c),(p,q,r),(x,y,z)|

Without, prove that : |{:(1+b,b+c,c+a),(p+q,q+r,r+p),(x+y,y+z,z+x),:}|=2|{:(a,b,c),(p,q,r),(x,y,z):}|

If A=|[a, b, c],[ x, y, z],[ p, q, r]| and B=|[q, -b, y],[ -p, a, -x],[ r,-c, z]| , without expanding or evaluating A and B , show that A+B=0 .