`A = [{:(e^(t), e^(-t)"cos"t, e^(-t)"sin"t),(e^(t),-e^(-t)"cos"t-e^(-t)"sin"t, -e^(-t)"sin"t + e^(-t)"cos"t),(e^(t), 2e^(-t)"sin"t, -2e^(-t)"cos"t):}]"then A is"`

Options

A. Invertible for all `t in R`
B. Invertible only if `t = pi/2`
C. Not Invertible for all `t in R`
D. Invertible only if `t = pi`

Answer

A

Explanation

`A=[(e^(t),e^(-t)cost,e^(-t)sint),(e^(t)-e^(-t),cost-e^(-t)sint-e^(-t),siny+e^(-t)cost),(e^(t),2e^(-t)sint,-2e^(-t)cost)]`
`det(A)=|A|=|(e^(t),e^(-t)cost,e^(-t)sint),(e^(t)-e^(-t),cos-e^(-t)sint,-e^(-t)sint+e^(-t)cost),(e^(t),2e^(-t)sint,-2e^(-t)cost)|`
`=e^(-t)|(1, cost,sint),(1,-cost-sint,-sint+cost),(1, 2sint,-2cost)|=e^(-t)|(1,cost, sint),(0,-2cost-sint,-2sint+cost),(0, 2sint-cost,-2cost-sint)|`
`=e^(-t)((2cost+sint)^(2)+(2sint-cost)^(2))gt 00 AA t.`