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Question
Using the properties of determinants & without expanding ⎡⎢⎣b+cq+ry+zc+ar+pz+xa+bp+qx+y⎤⎥⎦=2⎡⎢⎣apxbqycrz⎤⎥⎦
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Solution
⎡⎢⎣b+cq+ry+zc+ar+pz+xa+bp+qx+y⎤⎥⎦=⎡⎢⎣b+cq+ry+za−bp−qx−ya−cp−rx−z⎤⎥⎦R2→R2−R1,R3→R3−R1=⎡⎢⎣2a2p2xa−bp−qx−ya−cp−rx−z⎤⎥⎦R1→R1+R2+R3Taking out common 2 from R1 =2∣∣
∣∣apxa−bp−qx−ya−cp−rx−z∣∣
∣∣=2∣∣
∣∣apx−b−q−y−c−r−z∣∣
∣∣R2→R2−R1,R3→R3−R1taking out -1 from R2 and R3=2∣∣
∣∣apxbqycrz∣∣
∣∣
=⎡⎢⎣b+cq+ry+za−bp−qx−ya−cp−rx−z⎤⎥⎦R2→R2−R1,R3→R3−R1
=⎡⎢⎣2a2p2xa−bp−qx−ya−cp−rx−z⎤⎥⎦R1→R1+R2+R3
Taking out common 2 from R1
=2∣∣
∣∣apxa−bp−qx−ya−cp−rx−z∣∣
∣∣
=2∣∣
∣∣apx−b−q−y−c−r−z∣∣
∣∣R2→R2−R1,R3→R3−R1
taking out -1 from R2 and R3
=2∣∣
∣∣apxbqycrz∣∣
∣∣
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