1.
Let M be an invertible matrix, and let lambda be an eigenvalue of M. Show that 1/lambda is an
eigenvalue of M-1.
Solution
is eigenvalue so we have
Mx=x for some nonzero vector x
By multiplying both sides with M -1
we get
M -1
Mx=M -1
x
x=M -1
x
Now is nonzero because if =0, then Mx=0, so M -1
Mx=x=0. But x was nonzero
so we can divide by
1/x=M -1
x which shows 1/ is an eigenvalue of M -1