How to solve this question Without expanding, show that c2bc1a2ca1b2ab1 - Maths - Determinants

Question

How to solve this question????

W i t h o u t   e x p a n d i n g ,   s h o w   t h
a t   c 2 b c 1 a 2 c a 1 b 2 a b 1      
            ·    
            b 2 c b - 1 c 2 a c - 1 a
2 b a - 1

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

L H S
=c2bc1a2ca1b2ab1Multiply R1 by a, R2 by b and R3 by cL
H S
=1abcac2bacaa2bcabbb2acbccTake abc common from C1L
H S =abcabccacaaabbbbcc⇒L H S =cacaaabbbbcc iR H S
=b2cb-1c2ac-1a2ba-1Take -1 common from C3⇒R
H S
=-1b2cb1c2ac1a2ba1Multiply R1 by a, R2 by b and R3 by c⇒R
H S
=-1ab2cababc2abcba2bccacTake abc common from C1⇒R
H S =-1babacbcbacacInterchange C1 and C3⇒R H S
=-1-1aabbbbccccaa=aabbbbccccaaInterchange R1 and R2R
H S =-1bbccaabbccaaInterchange R1 and R3⇒R H S
=-1-1cacaaabbbbcc⇒R H S =cacaaabbbbcc
iiBy i and iiL H S =R H S
 Hence Proved

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How to solve this question???? W i t h o u t e x p a n d i n g , s h o w t h a t c 2 b c 1 a 2 c a 1 b 2 a b 1 · b 2 c b - 1 c 2 a c - 1 a 2 b a - 1

How to solve this question????

$W i t h o u t e x p a n d i n g, s h o w t h a t |\begin{matrix} c^{2} b & c & 1 \\ a^{2} c & a & 1 \\ b^{2} a & b & 1 \end{matrix}| \cdot |\begin{matrix} b^{2} c & b & - 1 \\ c^{2} a & c & - 1 \\ a^{2} b & a & - 1 \end{matrix}|$