# How to Use the Cayley-Hamilton Theorem to Find the Inverse Matrix

## Problem 502

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.

## Solution.

To apply the Cayley-Hamilton theorem, we first determine the characteristic polynomial $p(t)$ of the matrix $A$.

Let $I$ be the $3\times 3$ identity matrix.

We have

\begin{align*}

p(t)&=\det(A-tI)\\

&=\begin{vmatrix}

7-t & 2 & -2 \\

-6 &-1-t &2 \\

6 & 2 & -1-t

\end{vmatrix}\\[6pt]
&=(7-t)\begin{vmatrix}

-1-t & 2\\

2& -1-t

\end{vmatrix}

-2\begin{vmatrix}

-6 & 2\\

6& -1-t

\end{vmatrix}+(-2)\begin{vmatrix}

-6 & -1-t\\

6& 2

\end{vmatrix}\\[6pt]
&\text{(by the first row cofactor expansion)}\\[6pt]
&=-t^3+5t^2-7t+3.

\end{align*}

(You may also use the rule of Sarrus to compute the $3\times 3$ determinant.)

Thus, we have obtained the characteristic polynomial

\[p(t)=-t^3+5t^2-7t+3\]
of the matrix $A$.

The Cayley-Hamilton theorem yields that

\[O=p(A)=-A^3+5A^2-7A+3I,\]
where $O$ is the $3\times 3$ zero matrix.

(Here, don’t forget to put the identity matrix $I$.)

Rearranging terms, we have

\begin{align*}

&A^3-5A^2+7A=3I\\[6pt]
&\Leftrightarrow A(A^2-5A+7I)=3I\\[6pt]
&\Leftrightarrow A\left(\frac{1}{3}(A^2-5A+7I)\right)=I.

\end{align*}

Similarly, we have

\[\left(\frac{1}{3}(A^2-5A+7I)\right)A=I.\]
It follows from these two equalities that the matrix

\[\frac{1}{3}(A^2-5A+7I)\]
is the inverse matrix of $A$.

Therefore, we have

\begin{align*}

A^{-1}&=\frac{1}{3}(A^2-5A+7I)\\[6pt]
&=\frac{1}{3}\left(\, \begin{bmatrix}

25 & 8 & -8 \\

-24 &-7 &8 \\

24 & 8 & -7

\end{bmatrix}-5\begin{bmatrix}

7 & 2 & -2 \\

-6 &-1 &2 \\

6 & 2 & -1

\end{bmatrix}+7\begin{bmatrix}

1 & 0 & 0 \\

0 &1 &0 \\

0 & 0 & 1

\end{bmatrix} \,\right)\\[6pt]
&=\frac{1}{3}\begin{bmatrix}

-3 & -2 & 2 \\

6 &5 &-2 \\

-6 & -2 & 5

\end{bmatrix}.

\end{align*}

In summary, the inverse matrix of $A$ is

\[A^{-1}=\frac{1}{3}\begin{bmatrix}

-3 & -2 & 2 \\

6 &5 &-2 \\

-6 & -2 & 5

\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 matrix

\[A=\begin{bmatrix}

1 & 1 & 2 \\

9 &2 &0 \\

5 & 0 & 3

\end{bmatrix}\] using the Cayley–Hamilton theorem.

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

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