# Find the Inverse Matrix Using the Cayley-Hamilton Theorem

## Problem 421

Find the inverse matrix of the matrix
$A=\begin{bmatrix} 1 & 1 & 2 \\ 9 &2 &0 \\ 5 & 0 & 3 \end{bmatrix}$ using the Cayley–Hamilton theorem.

## Solution.

To use the Cayley-Hamilton theorem, we first compute the characteristic polynomial $p(t)$ of the matrix $A$. We have
\begin{align*}
&p(t)=\det(A-tI)\\
&\begin{vmatrix}
1-t & 1 & 2 \\
9 &2-t &0 \\
5 & 0 & 3-t
\end{vmatrix}\6pt] &=(-1)^{3+1}5\begin{vmatrix} 1 & 2\\ 2-t& 0 \end{vmatrix}+(-1)^{3+2}\cdot 0 \begin{vmatrix} 1-t & 2\\ 9& 0 \end{vmatrix}+(-1)^{3+3}(3-t)\begin{vmatrix} 1-t & 1\\ 9& 2-t \end{vmatrix} \\[6pt] & \text{(by the 3rd row cofactor expansion)}\\ &=5(2t-4)+0+(3-t)\left(\, (1-t)(2-t)-9 \,\right)\\ &=-t^3+6t^2+8t-41. \end{align*} Then the Cayley-Hamilton theorem yields that p(A)=O, the zero matrix. That is, we have \begin{align*} O=p(A)=-A^3+6A^2+8A-41I. \end{align*} Thus, we have \[41I=-A^3+6A^2+8A=A(-A^2+6A+8I), or equivalently
$I=A\left(\, \frac{1}{41}(-A^2+6A+8I) \,\right).$ It follows that the inverse matrix is given by
$A^{-1}=\frac{1}{41}(-A^2+6A+8I).$

By a direct computation, we have
$A^2=\begin{bmatrix} 20 & 3 & 8 \\ 27 &13 &18 \\ 20 & 5 & 19 \end{bmatrix}$ and
\begin{align*}
-A^2+6A+8I&=-\begin{bmatrix}
20 & 3 & 8 \\
27 &13 &18 \\
20 & 5 & 19
\end{bmatrix}+6\begin{bmatrix}
1 & 1 & 2 \\
9 &2 &0 \\
5 & 0 & 3
\end{bmatrix}+8\begin{bmatrix}
1 & 0 & 0 \\
0 &1 &0 \\
0 & 0 & 1
\end{bmatrix}\6pt] &=\begin{bmatrix} -6 & 3 & 4 \\ 27 &7 &-18 \\ 10 & -5 & 7 \end{bmatrix}. \end{align*} Therefore the inverse matrix is \[A^{-1}=\frac{1}{41}\begin{bmatrix} -6 & 3 & 4 \\ 27 &7 &-18 \\ 10 & -5 & 7 \end{bmatrix}.

## More Exercise

Test whether you understand how to find the inverse matrix using the Cayley-Hamilton theorem by the next problem.

Problem. Find the inverse matrix of the $3\times 3$ matrix
$A=\begin{bmatrix} 7 & 2 & -2 \\ -6 &-1 &2 \\ 6 & 2 & -1 \end{bmatrix}$ using the Cayley-Hamilton theorem.

The solution is given in the post “How to use the Cayley-Hamilton Theorem to Find the Inverse Matrix“.

### More Problems about the Cayley-Hamilton Theorem

Problems about the Cayley-Hamilton theorem and their solutions are collected on the page:

The Cayley-Hamilton Theorem

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1. 07/07/2017

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