Verifying the Math on Eigenvalues

In this exercise you'll find the eigenvalues \(\lambda\) of a matrix \(A\), and show that they satisfy the property that the matrix \(\lambda I - A\) is not invertible, with determinant equal to zero.

This exercise is part of the course

Linear Algebra for Data Science in R

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Exercise instructions

For the matrix A with the following R output:

     [,1] [,2]
[1,]    1    2
[2,]    1    1

find both eigenvalues.

Show that, for each eigenvalue lambda (\(\lambda\)), that the determinant of \(\lambda * I - A\) is equal to zero and hence the matrix \(\lambda * I - A\) is not invertible.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Compute the eigenvalues of A and store in Lambda
Lambda <- eigen(___)

# Print eigenvalues
print(Lambda$values[___])
print(Lambda$values[___])

# Verify that these numbers satisfy the conditions of being an eigenvalue
det(Lambda$values[___]*diag(2) - A)
det(Lambda$values[2]*diag(___) - A)

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