Question

Let
|4 0 1|
A= |2 3 2|
|25 0 4| .
For each eigenvalue λ, find the rank of the matrix λI - A.

Answer 1

@phi @ganeshie8

Answer 2

replace the diagonal 4 3 4 with (4-L), 3-L , 4-L
then expand it as if you were finding the determinant. This gives you the "characteristic equation" that you solve to find L

Answer 3

Thanks. Uhm, how about the rank?

Answer 4

for each eigenvalue, you find the matrix λI - A
now do row reduction and count the number of pivots

Answer 5

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Answer 6

Thank youuu :D

Answer 7

if you expand using the middle column (it has 2 zeros, so it's convenient),
the characteristic equation is
(3-x)[ (4-x)^2 -25] = 0
or
(3-x)[ x^2 -8x +16 -25]= 0
(x-3) (x^2 -8x -9) = 0
we see that this will be zero if
(x-3)= 0
or
x^2 -8x-9 = 0