Find the characteristic polynomial of the matrix: \[A = \begin{bmatrix}3&2&1\\0&1&1\\1&2&0 \end{bmatrix}\]
\[det(A - xI_u)=0\]
\[det(\begin{bmatrix}3&2&1\\0&1&1\\1&2&0 \end{bmatrix} - \begin{bmatrix}x&0&0\\0&x&0\\0&0&x \end{bmatrix})\] \[det\begin{bmatrix}3-x&2&1\\0&1-x&1\\1&2&-x \end{bmatrix}\] \[Rule\:of\:Sarrus = det(M) = det\begin{bmatrix}a11&a12&a13\\a21&a22&a23\\a31&a32&a33\end{bmatrix}=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-+a_{31}a_{22}a_{13}-+a_{32}a_{23}a_{11}+a_{33}a_{21}a_{12}\] \[\begin{bmatrix}3-x&2&1\\0&1-x&1\\1&2&-x \end{bmatrix}\begin{bmatrix}3-x&2\\0&1-x\\1&2&\end{bmatrix}\] \[(3-x)(1-x)(-x)+2-2(3-x)-(1-x)\] \[(3-3x-x+x^2)(-x)+2-6+2x-1+x\] \[-3x+3x^2+x^2-x^3-4+2x-1+x\] \[-3x+4x^2-x^3-5+3x\] \[p_A(x)=4x^2-x^3-5\]
matrix_A <- base::matrix(c(3,0,1,2,1,2,1,1,0),nrow=3,ncol=3)
(pracma::charpoly(matrix_A, info=T))## Error term: 0## $cp
## [1] 1 -4 0 5
##
## $det
## [1] -5
##
## $inv
## [,1] [,2] [,3]
## [1,] 0.4 -0.4 -0.2
## [2,] -0.2 0.2 0.6
## [3,] 0.2 0.8 -0.6