Question

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True or false: The matrix -4 1 0 1 4 0 0 1 is diagonalizablc.

Answer 1

we equation know that, a matroise is digonalizable if and only if algebric Multiplicity and Geometrie multiplicity for each eigen value are equal Algebraic multiplicity of an eigen value is the number of times it appears as the root of the characteristic Geometric multiplicity of an eigen valve is defined to the dimension of its eigen space. ne Geometric multiplicity of an eigen valve a for an nxn matrin A, is GEM (2) n-rank (Andf) Given mafria is A= 1-4 4 0 characteristic Polynomial of A is det (A-41) = la 4 0 0 0 lan 1,1,1 (1-x)3 in eigen values of A are Algebraic :: Algebie, multi plicuty of dal is an eigen value 3 Now, A-1 -41 04 0 0 : . rank (A-1)= 2 : Geometrie multiplicity of an eigen value dal is

GEM (d=1) = 3-rank (A-I) 3-2 :: Algebraic multiplicity and geometrie Multiplicity of an eigen value dal is not equal. Them by (o) the given matrix is not digonalizables :. the answer is false
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