# 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.

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## 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“.

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