- #1
Bertrandkis
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Question 1
Let A be an nxn matrix such that [tex](A-I)^{2}=O[/tex] where [tex]O[/tex] is the zero matrix
Prove that if [tex]{\lambda}[/tex] is an eigen value of A then [tex]{\lambda}=1[/tex]
My attempt
If [tex](A-I)^{2}=O[/tex] then [tex]A=I[/tex] (1)
if [tex]{\lambda}[/tex] is an eigen value of A then [tex]Ax={\lambda}x[/tex] (2)
replace (1) in (2) [tex]Ix={\lambda}x[/tex] , but [tex]Ix=x[/tex] therefore [tex]{\lambda}=1[/tex]
Question 2
If A is an orthogonal Matrix, then prove that det(A)=+-1
My attempt
if A is orthogonal then [tex]AA^{T}=I[/tex] and [tex]A^{-1}=A^{T}[/tex]
therefore [tex]AA^{-1}=I[/tex] and [tex]det(AA^{-1})=1[/tex] . Where does the + - comes from?
Let A be an nxn matrix such that [tex](A-I)^{2}=O[/tex] where [tex]O[/tex] is the zero matrix
Prove that if [tex]{\lambda}[/tex] is an eigen value of A then [tex]{\lambda}=1[/tex]
My attempt
If [tex](A-I)^{2}=O[/tex] then [tex]A=I[/tex] (1)
if [tex]{\lambda}[/tex] is an eigen value of A then [tex]Ax={\lambda}x[/tex] (2)
replace (1) in (2) [tex]Ix={\lambda}x[/tex] , but [tex]Ix=x[/tex] therefore [tex]{\lambda}=1[/tex]
Question 2
If A is an orthogonal Matrix, then prove that det(A)=+-1
My attempt
if A is orthogonal then [tex]AA^{T}=I[/tex] and [tex]A^{-1}=A^{T}[/tex]
therefore [tex]AA^{-1}=I[/tex] and [tex]det(AA^{-1})=1[/tex] . Where does the + - comes from?