Using properties of determinants, prove that ⎡⎢⎣b+cq+ry+zc+ar+pz+xa+bp+qx+y⎤⎥⎦=2⎡⎢⎣apxbqycrz⎤⎥⎦

First take LHS:
Δ=∣∣∣∣b+cq+ry+zc+ar+pz+xa+bp+qx+y∣∣∣∣
Applying,R1↔R3andR3↔R2, we get=∣∣∣∣a+bp+qx+yb+cq+ry+zc+ar+pz+x∣∣∣∣
Applying,R1→R1+R2+R3, we get=∣∣∣∣∣2(a+b+c)2(p+q+r)2(x+y+zb+cq+ry+zc+ar+pz+x∣∣∣∣∣⇒2∣∣∣∣a+b+cp+q+rx+y+zb+cq+ry+zc+ar+pz+x∣∣∣∣
Applying,R1→R1−R2=2∣∣∣∣apxb+cq+ry+zc+ar+pz+x∣∣∣∣
Applying,R3→R3−R1=2∣∣∣∣apxb+cq+ry+zcrz∣∣∣∣
Applying,R2→R2−R3
Δ=2∣∣∣∣apxbqycrz∣∣∣∣=RHSHence proved.