- #1
skrat
- 748
- 8
Using [itex]A\widetilde{A}=(detA)I[/itex] prove that [itex]\widetilde{A}[/itex] is reversible matrix if and only if [itex]A[/itex] is reversible. Also prove [itex]det(\widetilde{A})=(detA)^{n-1}[/itex] for any square matrix [itex]A[/itex].
First part:
1. direction:
Lets say [itex]\widetilde{A}[/itex] is reversible, this means that [itex]\widetilde{A}\widetilde{A}^{-1}=I[/itex] and [itex]det\widetilde{A}\neq 0[/itex].
[itex]A\widetilde{A}=(detA)I[/itex] can than be written as:
[itex]A=(detA)\widetilde{A}^{-1}[/itex] but now I don't know what else I can do here to prove that A is reversible?
First part:
1. direction:
Lets say [itex]\widetilde{A}[/itex] is reversible, this means that [itex]\widetilde{A}\widetilde{A}^{-1}=I[/itex] and [itex]det\widetilde{A}\neq 0[/itex].
[itex]A\widetilde{A}=(detA)I[/itex] can than be written as:
[itex]A=(detA)\widetilde{A}^{-1}[/itex] but now I don't know what else I can do here to prove that A is reversible?