Question

The Caley-Hamilton theorem states that "A matrix satisfies its own characteristic equation." Verify this theorem for the matrix $\mathit{M}$in equation (11.1). Hint: Substitute the matrix$\mathit{M}$forrole="math" localid="1658822242352" $\mathit{\lambda }$in the characteristic equation (11.4) and verify that you have a correct matrix equation. Further hint: Don't do all the arithmetic. Use (11.36) to write the left side of your equation as$\mathit{C}\left(D^2-7D+6\right){\mathit{C}}^{\mathbf{-}\mathbf{1}}$and show that the parenthesis$\mathbf{=}\mathbf{0}$. Remember that, by definition, the eigenvalues satisfy the characteristic equation.

Short answer

The matrices satisfy its own characteristic equation.

Step-by-step solution

Given information

The given matrix is$M=\left(\begin{array}{cc}5& -2\\ -2& 5\end{array}\right)$

Eigen values

Eigen values are the special set of scalar values that is associated with the set of linear equations most probably in the matrix equations.

Eigen values and inverse of the given matrix

Show that the matrix

$M=\left(\begin{array}{cc}5& -2\\ -2& 5\end{array}\right)$

satisfies its own characteristic equation

${\lambda }^2-7\lambda +6=0$

The eigenvalues are 1 and 6, the matrix$C$ is

$C=\frac{1}{\sqrt{5}}\left(\begin{array}{cc}1& -2\\ 2& 1\end{array}\right)$

and its inverse is

$C^{-1}=\frac{1}{\sqrt{5}}\left(\begin{array}{cc}1& -2\\ -2& 1\end{array}\right)$

Verify the characteristic equation

Verify that the characteristic equation is solved by the matrix $M$by plugging the matrix into the equation and using the property

$$\begin{gathered} M^n=C{D}^n{C}^{-1} \\ hereDis \\ D=\left(\begin{array}{cc}1& 0\\ 0& 6\end{array}\right) \end{gathered}$$

here is

Therefore,

$\mathrm{C}\left({\mathrm{D}}^2-7\mathrm{D}+6\mathrm{I}\right){\mathrm{C}}^{-1}=0$

Observe the bracket

$\left(\begin{array}{cc}{1}^2-7.1+6.1& 0\\ 0& 1^2-7.1+6.1\end{array}\right)=0$

which is expected because the eigenvalues satisfy the characteristic equation.