Discussion3

C22 Without using a calculator, find the eigenvalues of the matrix B.

\[ B = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \]

Answer:

\[ det(A- I) = det\begin{bmatrix} 2-\lambda & -1 \\ 1 & 1-\lambda \\ \end{bmatrix} = (2-\lambda)(1-\lambda)-(1)(-1) \]

\[ Characteristic\ polynomial = \lambda^2-3\lambda+3 \]

\[ Eigenvalues:\\ \lambda=\frac{3}{2} -\frac{\sqrt{3}i}{2} \\ \lambda=\frac{3}{2} +\frac{\sqrt{3}i}{2} \]

B <- matrix(c(2, -1, 1, 1), nrow = 2, byrow = TRUE)
B

##      [,1] [,2]
## [1,]    2   -1
## [2,]    1    1

result <- eigen(B)
eigenvalues <- result$values
cat("Eigenvalues:\n")

## Eigenvalues:

print(eigenvalues)

## [1] 1.5+0.866025i 1.5-0.866025i