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

Cayley-Hamilton Theorem Problems and Solutions

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.

 
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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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  1. 07/07/2017

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

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