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Shelnutt2
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Homework Statement
Find all n x n-matrices A such that Q-1 A Q = A for all invertible n x n matrices Q.
Homework Equations
None
The Attempt at a Solution
Proof by induction
Assumption: True for A = k(In) where k is a real number
Base case: k = 1
When k =1
A = 1 * In = In
Q-1 * In * Q = In
(Q-1 * In) * Q = In
(Q-1) * Q =In
In = In.
When k = m
A = m * In
Q-1 * (m *In) * Q = m * In
m * (Q-1 * In) * Q = m * In
m * (Q-1) * Q = m * In
m * In = m * In.
m * In + In = In * (m + 1) =
When k = m+1
A = m+1 * In
Q-1 * ((m + 1) *In) * Q = (m + 1) * In
(m + 1) * (Q-1 * In) * Q = (m + 1) * In
(m + 1) * (Q-1) * Q = (m + 1) * In
(m + 1) * In = (m + 1) * In.
Since they equal this is true by induction.
This is one of 6 bonus problems the linear algebra class were given. It's the first three people to turn in each of the problems gets the two points. However you only get one submission. So I thought I'd have someone else check my work to see if it is correct or not before I turn it in.
Thank you in advance!