Given, det(A)=2 det(B)=3 and det((det(5(detA)B))A2)=2a3b5c ⇒|det(5(detA)B)A2
|=2a3b5c ⇒||5(detA)B)A2∣=2a3b5c ⇒||5|A|B|A2
|=2a3b5c ⇒||5.2.B|A2
|=2a⋅3b⋅5c ⇒||10B|A2
|=2a⋅3b⋅5c ⇒|102⋅
|B|A2
|=2a⋅3b⋅5c As |k.A|=kn|A| ⇒|100×3A2
|=2a⋅3b⋅5c ⇒(300)2⋅|A2
|=2a⋅3b⋅5c ⇒(300)2⋅22=2a⋅3b⋅5c ⇒9×100×100×22=2a⋅3b⋅5c ⇒32×22×52×22×52×22=2a⋅3b⋅5c ⇒26⋅32⋅54=2a⋅3b⋅5c Comparing both sides, we get a=6,b=2,c=4 ∴a+b+c=6+2+4=12