= Solution
A <linear endomorphism> is <diagonalizable> over its <field> precisely when its <minimal polynomial> is a product of \b[distinct <linear polynomials> over that same <field>]. Splitting alone is insufficient: repeated factors permit nontrivial <Jordan blocks>.
For the displayed <matrix>, a <determinant> expansion gives
$$
\chi_A(x)=x^3+3x^2+6x-10=(x-1)(x^2+4x+10).
$$
These formulas can be reduced in each requested <characteristic>. The four cases below distinguish failure to split from failure to have distinct factors.
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