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Question Number 115508 by bemath last updated on 26/Sep/20
If determinant (((a a^2 1+a^3 )),((b b^2 1+b^3 )),((c c^2 1+c^3 )))= 0 a≠b≠c → { ((a =?)),((b=? )),((c=?)) :}
If|aa21+a3bb21+b3cc21+c3|=0a≠b≠c→{a=?b=?c=?
Answered by bobhans last updated on 26/Sep/20
⇒ determinant (((a a^2 1)),((b b^2 1)),((c c^2 1)))+ determinant (((a a^2 a^3 )),((b b^2 b^3 )),((c c^2 c^3 )))= 0 ⇒(−1) determinant (((1 a^2 a)),((1 b^2 b)),((1 c^2 c)))+ abc determinant (((1 a a^2 )),((1 b b^2 )),((1 c c^2 )))= 0 ⇒(−1)^2 determinant (((1 a a^2 )),((1 b b^2 )),((1 c c^2 )))+ abc determinant (((1 a a^2 )),((1 b b^2 )),((1 c c^2 )))= 0 ⇒(1+abc) determinant (((1 a a^2 )),((1 b b^2 )),((1 c c^2 )))= 0 ⇒(1+abc) determinant (((1 a a^2 )),((0 b−a b^2 −a^2 )),((0 c−a c^2 −a^2 )))= 0 ⇒(1+abc) determinant (((b−a b^2 −a^2 )),((c−a c^2 −a^2 )))= 0 ⇒(1+abc)[(b−a)(c−a)(c+a)−(c−a)(b−a)(b+a)]=0 ⇒(1+abc)(b−a)(c−a) [ c+a−b−a] = 0 (1+abc)(b−a)(c−a)(c−b) = 0 since a≠b≠c ; so we get abc = −1 let a = k , b = l → c = −(1/(k.l))
⇒|aa21bb21cc21|+|aa2a3bb2b3cc2c3|=0⇒(−1)|1a2a1b2b1c2c|+abc|1aa21bb21cc2|=0⇒(−1)2|1aa21bb21cc2|+abc|1aa21bb21cc2|=0⇒(1+abc)|1aa21bb21cc2|=0⇒(1+abc)|1aa20b−ab2−a20c−ac2−a2|=0⇒(1+abc)|b−ab2−a2c−ac2−a2|=0⇒(1+abc)[(b−a)(c−a)(c+a)−(c−a)(b−a)(b+a)]=0⇒(1+abc)(b−a)(c−a)[c+a−b−a]=0(1+abc)(b−a)(c−a)(c−b)=0sincea≠b≠c;sowegetabc=−1leta=k,b=l→c=−1k.l
Commented by bemath last updated on 26/Sep/20
gave kudos
gavekudos
Answered by TANMAY PANACEA last updated on 26/Sep/20
determinant ((a,a^2 ,1),(b,b^2 ,1),(c,c^2 ,1))+ determinant ((a,a^2 ,a^3 ),(b,b^2 ,b^3 ),(c,c^2 ,c^3 ))=0 determinant ((a,a^2 ,1),(b,b^2 ,1),(c,c^2 ,1))+abc determinant ((1,a,a^2 ),(1,b,b^2 ),(1,c,c^2 ))=0 determinant ((1,a,a^2 ),(1,b,b^2 ),(1,c,c^2 ))+abc determinant ((1,a,a^2 ),(1,b,b^2 ),(1,c,c^2 ))=0 D+abc×D=0 D(1+abc)=0 D≠0 so abc=−1 let a=k b=(1/k) c=−1 i have just tried...
|aa21bb21cc21|+|aa2a3bb2b3cc2c3|=0|aa21bb21cc21|+abc|1aa21bb21cc2|=0|1aa21bb21cc2|+abc|1aa21bb21cc2|=0D+abc×D=0D(1+abc)=0D≠0soabc=−1leta=kb=1kc=−1ihavejusttried…
Commented by bemath last updated on 26/Sep/20
gave kudos sir.
gavekudossir.

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