If A = [et&et cos t&e-t sin t et&-et cos t -e-t sin t&-e-t sin t+ e-t cos t et&2e-t sin t&-2e-t cos t] Then A is -

Question

If A = [et&et cos t&e-t sin t et&-et cos t -e-t sin t&-e-t sin t+ e-t cos t et&2e-t sin t&-2e-t cos t] Then A is - (A) Invertible only if t = (π/2) (B) not invertible for any t ϵ R (C) invertible for all t ϵ R (D) invertible only if t = π

Tardigrade Admin · Accepted Answer

|A| = e-t |1& cos t& sin t 1&- cos t - sin t&- sin t + cos t 1&2 sin t&-2 cos t| = e-t [5 cos2 t + 5 sin2 t] ∀ t ∈ R = 5e-t ≠0 ∀ t ∈ R

Q. If $A = ⎣ ⎡ e^{t} e^{t} e^{t} e^{t} cos t - e^{t} cos t - e^{- t} sin t 2 e^{- t} sin t e^{- t} sin t - e^{- t} sin t + e^{- t} cos t - 2 e^{- t} cos t ⎦ ⎤$ Then $A$ is -

Invertible only if $t = \frac{π}{2}$

not invertible for any $t ϵ R$

invertible for all $t ϵ R$

invertible only if $t = π$

$∣ A ∣ = e^{- t} ∣ ∣ 111 cos t - cos t - sin t 2 sin t sin t - sin t + cos t - 2 cos t ∣ ∣$
$= e^{- t} [5 cos^{2} t + 5 sin^{2} t] \forall t \in R$
$= 5 e^{- t} \neq = 0\forall t \in R$

Q. If A=⎣⎡​etetet​etcost−etcost−e−tsint2e−tsint​e−tsint−e−tsint+e−tcost−2e−tcost​⎦⎤​ Then A is -

Invertible only if t=2π​

not invertible for any tϵR

invertible for all tϵR

invertible only if t=π