## 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.

of the day

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.

In this post, we study the **Fundamental Theorem of Finitely Generated Abelian Groups**, and as an application we solve the following problem.

**Problem**.

Let $G$ be a finite abelian group of order $n$.

If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic to the cyclic group $Z_n=\Zmod{n}$ of order $n$.

**(a)** Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$

**(b)** Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an eigenvalue.

If $M$ is a finite abelian group, then $M$ is naturally a $\Z$-module.

Can this action be extended to make $M$ into a $\Q$-module?

Add to solve later Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $I$ be an ideal of $R$.

Let $M’$ be the subset of elements $a$ of $M$ that are annihilated by some power $I^k$ of the ideal $I$, where the power $k$ may depend on $a$.

Prove that $M’$ is a submodule of $M$.

Let $R$ be a ring with $1$. Let $M$ be an $R$-module. Consider an ascending chain

\[N_1 \subset N_2 \subset \cdots\]
of submodules of $M$.

Prove that the union

\[\cup_{i=1}^{\infty} N_i\]
is a submodule of $M$.

**(a)** Let $R$ be a commutative ring. If we regard $R$ as a left $R$-module, then prove that any two distinct elements of the module $R$ are linearly dependent.

**(b)** Let $f: M\to M’$ be a left $R$-module homomorphism. Let $\{x_1, \dots, x_n\}$ be a subset in $M$. Prove that if the set $\{f(x_1), \dots, f(x_n)\}$ is linearly independent, then the set $\{x_1, \dots, x_n\}$ is also linearly independent.

Read solution

Let $R$ be a ring with $1$. Let

\[0\to M\xrightarrow{f} M’ \xrightarrow{g} M^{\prime\prime} \to 0 \tag{*}\]
be an exact sequence of left $R$-modules.

Prove that if $M$ and $M^{\prime\prime}$ are finitely generated, then $M’$ is also finitely generated.

Add to solve later Suppose that $f:R\to R’$ is a surjective ring homomorphism.

Prove that if $R$ is a Noetherian ring, then so is $R’$.

Let $f: R\to R’$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R’$.

Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.

Add to solve laterLet $f:R\to R’$ be a ring homomorphism. Let $I’$ be an ideal of $R’$ and let $I=f^{-1}(I)$ be the preimage of $I$ by $f$. Prove that $I$ is an ideal of the ring $R$.

Add to solve later Let $R$ be a ring with $1$ and let $M$ be a left $R$-module.

Let $S$ be a subset of $M$. The **annihilator** of $S$ in $R$ is the subset of the ring $R$ defined to be

\[\Ann_R(S)=\{ r\in R\mid rx=0 \text{ for all } x\in S\}.\]
(If $rx=0, r\in R, x\in S$, then we say $r$ **annihilates** $x$.)

Suppose that $N$ is a submodule of $M$. Then prove that the annihilator

\[\Ann_R(N)=\{ r\in R\mid rn=0 \text{ for all } n\in N\}\]
of $M$ in $R$ is a $2$-sided ideal of $R$.

Let $R$ be a ring with $1$. An element of the $R$-module $M$ is called a **torsion element** if $rm=0$ for some nonzero element $r\in R$.

The set of torsion elements is denoted

\[\Tor(M)=\{m \in M \mid rm=0 \text{ for some nonzero} r\in R\}.\]

**(a)** Prove that if $R$ is an integral domain, then $\Tor(M)$ is a submodule of $M$.

(Remark: an integral domain is a commutative ring by definition.) In this case the submodule $\Tor(M)$ is called **torsion submodule** of $M$.

**(b)** Find an example of a ring $R$ and an $R$-module $M$ such that $\Tor(M)$ is not a submodule.

**(c)** If $R$ has nonzero zero divisors, then show that every nonzero $R$-module has nonzero torsion element.

Let $R$ be a ring with $1$ and $M$ be a left $R$-module.

**(a)** Prove that $0_Rm=0_M$ for all $m \in M$.

Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$.

To simplify the notations, we ignore the subscripts and simply write

\[0m=0.\]
You must be able to and must judge which zero elements are used from the context.

**(b) **Prove that $r0=0$ for all $s\in R$. Here both zeros are $0_M$.

**(c)** Prove that $(-1)m=-m$ for all $m \in M$.

**(d)** Assume that $rm=0$ for some $r\in R$ and some nonzero element $m\in M$. Prove that $r$ does not have a left inverse.

Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix.

Prove that the matrix $A$ has at least one real eigenvalue.

Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that

\[\mathbf{y}A=\mathbf{y}.\]
(Here a row vector means a $1\times n$ matrix.)

Prove that there is a nonzero column vector $\mathbf{x}$ such that

\[A\mathbf{x}=\mathbf{x}.\]
(Here a column vector means an $n \times 1$ matrix.)

Recall that a complex matrix is called **Hermitian** if $A^*=A$, where $A^*=\bar{A}^{\trans}$.

Prove that every Hermitian matrix $A$ can be written as the sum

\[A=B+iC,\]
where $B$ is a real symmetric matrix and $C$ is a real skew-symmetric matrix.

Let $A$ be an $n\times n$ real matrix.

Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.

Add to solve later Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$.

Furthermore, suppose that

\[|\lambda_1| > |\lambda_2| \geq \cdots \geq |\lambda_n|.\]
Let

\[\mathbf{x}_0=c_1\mathbf{u}_1+c_2\mathbf{u}_2+\cdots+c_n\mathbf{u}_n\]
for some real numbers $c_1, c_2, \dots, c_n$ and $c_1\neq 0$.

Define

\[\mathbf{x}_{k+1}=A\mathbf{x}_k \text{ for } k=0, 1, 2,\dots\]
and let

\[\beta_k=\frac{\mathbf{x}_k\cdot \mathbf{x}_{k+1}}{\mathbf{x}_k \cdot \mathbf{x}_k}=\frac{\mathbf{x}_k^{\trans} \mathbf{x}_{k+1}}{\mathbf{x}_k^{\trans} \mathbf{x}_k}.\]

Prove that

\[\lim_{k\to \infty} \beta_k=\lambda_1.\]

Let $G$ be a group. Suppose that we have

\[(ab)^3=a^3b^3\]
for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$.

Then prove that $G$ is an abelian group.

Add to solve later