Find the Inverse eigenvalue ([[1,2,3],[4,5,6],[7,8,9]])

Problem

A=[[1,2,3],[4,5,6],[7,8,9]]

Solution

1. Identify the matrix A and the goal to find its eigenvalues. The term "inverse eigenvalue" typically refers to the eigenvalues of the inverse matrix A(−1) which are the reciprocals of the eigenvalues of A

2. Calculate the determinant of matrix A to check for invertibility.

det(A)=1*(5⋅9−6⋅8)−2*(4⋅9−6⋅7)+3*(4⋅8−5⋅7)

det(A)=1*(45−48)−2*(36−42)+3*(32−35)

det(A)=−3−2*(−6)+3*(−3)

det(A)=−3+12−9=0

3. Conclude that because the determinant is zero, the matrix A is singular (not invertible).

4. Determine the eigenvalues of A to see if an "inverse eigenvalue" exists. Since the matrix is singular, at least one eigenvalue λ is equal to 0

5. Observe that the reciprocal of a zero eigenvalue is undefined. Therefore, the inverse eigenvalues do not exist because the matrix A(−1) does not exist.

Final Answer

The inverse eigenvalues do not exist because *det(A)=0

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