Discussion3

Problem C10 page 388

Find the charactertistic polynomial of the matrix

library(knitr)
library(pracma)

A = matrix(c (1,2,3,4), nrow=2, byrow=T)
print(A)

##      [,1] [,2]
## [1,]    1    2
## [2,]    3    4

Solution using R

charpoly(A)

## [1]  1 -5 -2

This gives us the characteristic polynomial as below:

\[{ PA(x)=λ}^{ 2 }-5{ λ } -2\]

Solution without using charpoly

We will find the characteristic polynomial of a 2X2 matrix

\[A=\begin{vmatrix} 1 & 2 \\ 3 & 4\end{vmatrix}\]

We must solve the following to get eigenvalues

\[det(\begin{vmatrix} λ & 0 \\ 0 & λ \end{vmatrix}-\begin{vmatrix} 1 & 2\\ 3 & 4\end{vmatrix})\ =0\]

Subtracting gives us

\[det(\begin{vmatrix} λ-1 & -2 \\ -3 & λ-4\end{vmatrix})= 0\]

Next step is to solve for the determinant using det(A)=ad-bc

\[(λ-1)(λ-4)- (-3)(-2)=0\]

\[λ(λ-4)-1(λ-4)-6=0\] \[λ^2-4λ-λ+4-6=0\] \[λ^2-5λ-2=0\]

This is the same equation as the one given by the coordinated from charpoly function.