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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