Prove the identity without evaluating the determinants

[a1+b1t a2+b2t a3+b3t] [a1 a2 a3]

[a1t + b1 a2t+b2 a3t+b3] = (1-t^2) [b1 b2 b3]

[c1 c2 c3 ] [c1 c2 c3]

I am not even sure where to begin with this as I do not know the method required to prove the identity.

Any help would greatly appreciated.

Thanks

I'm just editing this to include a picture that should make the problem clearer (also, it's t^2, not t2 in the scalar before the matrix on the RHS of the equation):