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]| `

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)|

Show that: |b+cc+a a+b q+r r+p p+q y+z z+xx+y|=2|a b c p q r x y z|

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

If Delta_1=|[a, b, c],[ x, y, z],[ p, q, r]| and Delta_2=|[q,-b, y],[-p, a,-x],[ r,-c, z]|, without expending or evaluating Delta_1 and Delta_2 , show that Delta_1+Delta_2=0

Prove that: |{:(a, b, c), (x, y, z), (p, q, r):}|=|{:(y, b, q), (x, a, p), (z, c, r):}|

If Delta_(1)=|[a,b, c],[x, y, z],[p,q ,r]|"and "Delta_(2) |[q,-b, y],[-p, a, -x],[r,-c ,z]| then without expanding Delta_(1) " and "Delta_(2), "prove that "Delta_(1) + Delta_(2) =0

Using properties of determinants, prove that |(b+c,q+r,y+z),(c+a,r+p,z+x),(c+b,p+q,x+y)|=2|(a,p,x),(b,q,y),(c,r,z)|

Use the properties of determinant 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 expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

If |[a, p, x],[ b, q, y], [c, r, z]|=16 , then the value of |[p+x, a+x, a+p], [q+y ,b+y, b+q],[ x+z, c+z, c+r]| is 4 (b) 8 (c) 16 (d) 32