Since given matrix A is invertible ⇒∣A∣=0
∣A∣=e−t∣111cost−cost−sint2sintsint−sint+cost−2cost∣
R2→R2−R1
R3→R3−R1
=e−t∣100cost−2cost−sint2sint−costsint−2sint+cost−2cost−sint∣
=e−t(2cost+sint)2+(2sint−cost)2
=5e−t
∣A∣=5e−t=0∀t∈R
Hence, given matrix is always invertible.