Answer to Question #254843 in Linear Algebra for Sabelo Xulu

Question #254843

Suppose A is a matrix with characteristic polynomial p(λ\lambda ) =λ\lambda3 - λ\lambda

a) What is the order of the matrix A?

b) Is A invertible?

c) Is A diagonalisable?

d) Find the eigenvalues of A2

1
Expert's answer
2021-10-25T15:16:15-0400

a) The characteristics

polynomial P(λ\lambda ) has degree n

\therefore A is of order 3×33×3


b)The roots of characteristic polynomial

λ\lambda 3 -λ=0\lambda=0 are

λ(λ\lambda(\lambda 21)=0-1)=0

    λ(λ1)(λ+1)=0\implies \lambda(\lambda-1)(\lambda+1)=0

λ=0,\therefore \lambda=0, or 1-1 or 11


A matrix is invertible     \iff

det(A)/=det(A) \mathrlap{\,/}{=}

det A is exactly the product of A eigenvalues

0.1.1=00.-1.1=0

Since the product is

Then AA is not invertible


c) A is diagonalisable because the characteristic polynomial can be factored into distinct linear factors

I.e λ\lambda 3 - λ=λ(λ\lambda=\lambda(\lambda 2 -11 )

=λ(λ1)(λ+1)\lambda (\lambda-1)(\lambda+1)


d) Eigenvalues of AA 2 are (0)(0) 2,(1)(-1) 2 and (1)(1) 2

Which are

0,10,1 and 11








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