Find one solution to this system and Find the general soluti

Find one solution to this system, and Find the general solution to the linear system

Let A = (1 -4 0 0 -6 2 1 -10 2) (1 0 -4/3 0 1 -1/3 0 0 0). Consider the linear system (1 -4 0 0 -6 2 1 -10 2) (x_1 x_2 x_3) = (-1 2 1). Find one solution to this system. Find the general solution to the linear system in , expressed as a vector plus a vector space.

Solution

can you please check those 2 matrices?

I calculated their eigenvalues and they are not coming out to be same, which is bothering me a lot. Since, similar matrices should have same eigenvalues.

I calculated by hand and even online, i am getting different eigenvaues for these matrices.

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Idea to solve such problems:

A~B implies A = PBP^-1 . so,

AX =b imlpies - B x = P^-1bP and B is simple matrix and then system can be solved.

OR

Find linear trasformation and 2 basis set which are giving us 2 similar matrices and once we have that, we can find change of basis matrix.