Confirm the identities without evaluating the determinants directly

Confirm the identities without evaluating the determinants directly.

[a1 + b1t a2 + b2t a3 + b3t, a1t + b1 a2t + b2 a3t + b3, c1 c2 c3] = (1 - t2) [a1 a2 a3, b1 b2 b3, c1 c2 c3]