Question

Consider the matrix

a. a = 4 0 0 1 3 0 −4 5 −1 find the characteristic polynomial for the matrix
a. (write your answer in terms of λ.)

Answer 1

`\mathbf A=\begin{bmatrix}4&0&0\\1&3&0\\-4&5&-1\end{bmatrix}`

The characteristic polynomial is given by
`\det(\mathbf A-\lambda\mathbf I)`:

`\mathbf A-\lambda\mathbf I=\begin{bmatrix}4-\lambda&0&0\\1&3-\lambda&0\\-4&5&-1-\lambda\end{bmatrix}`

Compute the determinant by Laplace expansion along the first row:

`\det(\mathbf A-\lambda\mathbf I)=(4-\lambda)\begin{vmatrix}3-\lambda&0\\5&-1-\lambda\end{vmatrix}=(4-\lambda)(3-\lambda)(-1-\lambda)`

`\implies\det(\mathbf A-\lambda\mathbf I)=-\lambda^3+6\lambda^2-5\lambda-12`