## Prove a Group is Abelian if $(ab)^2=a^2b^2$

## Problem 401

Let $G$ be a group. Suppose that

\[(ab)^2=a^2b^2\]
for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.

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Let $G$ be a group. Suppose that

\[(ab)^2=a^2b^2\]
for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.

Find all the eigenvalues and eigenvectors of the matrix

\[A=\begin{bmatrix}

10001 & 3 & 5 & 7 &9 & 11 \\

1 & 10003 & 5 & 7 & 9 & 11 \\

1 & 3 & 10005 & 7 & 9 & 11 \\

1 & 3 & 5 & 10007 & 9 & 11 \\

1 &3 & 5 & 7 & 10009 & 11 \\

1 &3 & 5 & 7 & 9 & 10011

\end{bmatrix}.\]

(*MIT, Linear Algebra Homework Problem*)

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Prove that the cubic polynomial $x^3-2$ is irreducible over the field $\Q(i)$.

Add to solve laterSuppose $A$ is a positive definite symmetric $n\times n$ matrix.

**(a)** Prove that $A$ is invertible.

**(b)** Prove that $A^{-1}$ is symmetric.

**(c)** Prove that $A^{-1}$ is positive-definite.

(*MIT, Linear Algebra Exam Problem*)

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A real symmetric $n \times n$ matrix $A$ is called **positive definite** if

\[\mathbf{x}^{\trans}A\mathbf{x}>0\]
for all nonzero vectors $\mathbf{x}$ in $\R^n$.

**(a)** Prove that the eigenvalues of a real symmetric positive-definite matrix $A$ are all positive.

**(b)** Prove that if eigenvalues of a real symmetric matrix $A$ are all positive, then $A$ is positive-definite.

Suppose that the vectors

\[\mathbf{v}_1=\begin{bmatrix}

-2 \\

1 \\

0 \\

0 \\

0

\end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix}

-4 \\

0 \\

-3 \\

-2 \\

1

\end{bmatrix}\]
are a basis vectors for the null space of a $4\times 5$ matrix $A$. Find a vector $\mathbf{x}$ such that

\[\mathbf{x}\neq0, \quad \mathbf{x}\neq \mathbf{v}_1, \quad \mathbf{x}\neq \mathbf{v}_2,\]
and

\[A\mathbf{x}=\mathbf{0}.\]

(Stanford University, Linear Algebra exam problem)

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Determine the values of $x$ so that the matrix

\[A=\begin{bmatrix}

1 & 1 & x \\

1 &x &x \\

x & x & x

\end{bmatrix}\]
is invertible.

For those values of $x$, find the inverse matrix $A^{-1}$.

**(a)** Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as

\[A=BC,\]
where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.

Prove that the matrix $A$ cannot be invertible.

**(b)** Let $A$ be a $2\times 2$ matrix and suppose that $A$ can be written as

\[A=BC,\]
where $B$ is a $ 2\times 3$ matrix and $C$ is a $3\times 2$ matrix.

Can the matrix $A$ be invertible?

Add to solve later Let $V$ be the subspace of $\R^4$ defined by the equation

\[x_1-x_2+2x_3+6x_4=0.\]
Find a linear transformation $T$ from $\R^3$ to $\R^4$ such that the null space $\calN(T)=\{\mathbf{0}\}$ and the range $\calR(T)=V$. Describe $T$ by its matrix $A$.

**(a)** Is the matrix $A=\begin{bmatrix}

1 & 2\\

0& 3

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

3 & 0\\

1& 2

\end{bmatrix}$?

**(b)** Is the matrix $A=\begin{bmatrix}

0 & 1\\

5& 3

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

1 & 2\\

4& 3

\end{bmatrix}$?

**(c)** Is the matrix $A=\begin{bmatrix}

-1 & 6\\

-2& 6

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

3 & 0\\

0& 2

\end{bmatrix}$?

**(d)** Is the matrix $A=\begin{bmatrix}

-1 & 6\\

-2& 6

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

1 & 2\\

-1& 4

\end{bmatrix}$?

Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.

Add to solve later**(a)** A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.

Find $\det(A)$.

**(b)** A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.

**(c)** A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of $A$?

(Harvard University, Linear Algebra exam problem)

Add to solve laterLet $A$ be $n\times n$ matrix and let $\lambda_1, \lambda_2, \dots, \lambda_n$ be all the eigenvalues of $A$. (Some of them may be the same.)

For each positive integer $k$, prove that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of $A^k$.

Add to solve laterLet $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities.

What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?

(*The Ohio State University, Linear Algebra Final Exam Problem*)

Find all eigenvalues of the matrix

\[A=\begin{bmatrix}

0 & i & i & i \\

i &0 & i & i \\

i & i & 0 & i \\

i & i & i & 0

\end{bmatrix},\]
where $i=\sqrt{-1}$. For each eigenvalue of $A$, determine its algebraic multiplicity and geometric multiplicity.

Let

\[A=\begin{bmatrix}

2 & -1 & -1 \\

-1 &2 &-1 \\

-1 & -1 & 2

\end{bmatrix}.\]
Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$.

That is, find a nonsingular matrix $A$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

Let $A$ be an $n\times n$ matrix with the characteristic polynomial

\[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\]
Assume that the matrix $A$ is diagonalizable.

**(a)** Find the size of the matrix $A$.

**(b)** Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue $\lambda=2$.

**(c)** Find the nullity of $A$.

(The Ohio State University, Linear Algebra final exam problem)

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Let

\[A=\begin{bmatrix}

1 & 1 & 1 \\

0 &0 &1 \\

0 & 0 & 1

\end{bmatrix}\]
be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$.

Let $\lambda$ be an eigenvalue of $n\times n$ matrices $A$ and $B$ corresponding to the same eigenvector $\mathbf{x}$.

**(a)** Show that $2\lambda$ is an eigenvalue of $A+B$ corresponding to $\mathbf{x}$.

**(b)** Show that $\lambda^2$ is an eigenvalue of $AB$ corresponding to $\mathbf{x}$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

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