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A is 5×5 matrix, all of whose entries are 1, then
(a) A is not diagonalizable
(b) A is idempotent
(c) A is nilpotent
(d) The minimal polynomial and the characteristics polynomial of A are not equal
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A

is symmetric (or selfadjoint, if your matrices are complex), so it is diagonalizable.

It is not idempotent, because A2=5A

.

It is not nilpotent, because An=5n−1A

.

The minimal polynomial of A
is pm(t)=t(t−5), while the characteristic polynomial is pc(t)=t4(t−5)

. So they are different.

The characteristic polynomial can be obtained this way: we have
A=⎡⎣⎢⎢1⋮10⋮0⋯⋱⋯0⋮0⎤⎦⎥⎥⎡⎣⎢⎢⎢⎢⎢100⋯⋯⋱⋯100⎤⎦⎥⎥⎥⎥⎥,
so its spectrum agrees with that of
⎡⎣⎢⎢⎢⎢⎢100⋯⋯⋱⋯100⎤⎦⎥⎥⎥⎥⎥⎡⎣⎢⎢1⋮10⋮0⋯⋱⋯0⋮0⎤⎦⎥⎥=⎡⎣⎢⎢⎢⎢⎢50000⋯⋯⋱⋯000⎤⎦⎥⎥⎥⎥⎥.
The eigenvalues are then 5,0,0,0,0, and so pc(t)=t4(t−5). As A5−5A=0, the minimal polynomial is at most t(t−5). But it also requires 0 and 5 as roots, so pm(t)=t(t−5).

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