# Introduction to Matrices

## Introduction to Matrices

Definition

1. The trace $\tr(A)$ of an $n\times n$ matrix $A=(a_{ij})$ is the sum of the diagonal entries of $A$. That is, $\tr(A)=\sum_{i=1}^n a_{ii}$.
2. The transpose $A^{\trans}$ of an $m\times n$ matrix $A$ is the $n\times m$ matrix whose $(i,j)$-entry is $a_{j i}$.
3. A matrix $A$ is called symmetric if $A^{\trans}=A$.
4. We say two matrices $A, B$ commute if $AB=BA$.
5. The entries $a_{ii}$ of a matrix $A=(A_{ij})$ are called diagonal entries.
6. A diagonal matrix is a square matrix whose non-diagonal entries are all zero.
7. An $n\times n$ matrix whose diagonal entries are all $1$ is called the identity matrix and denoted by $I_n$, or simply by $I$.
Summary

1. For matrices $A, B$, and a scalar $r$, the followings are true when the expressions defined:
1. $(A+B)^{\trans}=A^{\trans}+B^{\trans}$
2. $(AB)^{\trans}=B^{\trans}A^{\trans}$
3. $(rA)^{\trans}=rA^{\trans}$
2. For $n\times n$ matrices $A, B$, and a scalar $r$, we have
1. $\tr(A+B)=\tr(A)+\tr(B)$
2. $\tr(rA)=r\tr(A)$

=solution

### Problems

1. Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix. Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a counterexample.

2. Let $A$ and $B$ be $2\times 2$ matrices. Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2-B^2$.

3. Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix. Is it true that the matrix product with opposite order $BA$ is also the zero matrix? If so, give a proof. If not, give a counterexample.

4. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix.

5. Let
$A=\begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix} 1\\ 0 \end{bmatrix}.$ Compute $A^{2017}\mathbf{u}$.
(The Ohio State University)

6. 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 $n$.

7. A square matrix $A$ is called idempotent if $A^2=A$.
(a) Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be the $n\times n$ identity matrix. Prove that the matrix $I-A$ is an idempotent matrix.
(b) Assume that $A$ is an $n\times n$ nonzero idempotent matrix. Then determine all integers $k$ such that the matrix $I-kA$ is idempotent.
(c) Let $A$ and $B$ be $n\times n$ matrices satisfying $AB=A$ and $BA=B$. Then prove that $A$ is an idempotent matrix.

8. Let $A=\begin{bmatrix} 1 & 3\\ 2& 4 \end{bmatrix}.$
(a) Find all matrices $B=\begin{bmatrix} x & y\\ z& w \end{bmatrix}$ such that $AB=BA$.
(b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that $AB=BA$ and $AC \neq CA$.

9. Let
$A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &1 &0 \end{bmatrix}, C=\begin{bmatrix} 1 & 2\\ 0& 6 \end{bmatrix}, \mathbf{v}=\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}.$ Then compute and simplify the following expression.
$\mathbf{v}^{\trans}\left( A^{\trans}-(A-B)^{\trans}\right)C.$
10. Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$. Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.
11. Prove the following identity for any positive integer $n$.
$\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}^n=\begin{bmatrix} \cos n\theta & -\sin n\theta\\ \sin n\theta& \cos n\theta \end{bmatrix}.$
12. Let $A$ and $B$ be $n \times n$ matrices. Is it always true that $\tr (A B) = \tr (A) \tr (B)$? If it is true, prove it. If not, give a counterexample.
13. Let $A$ be an $n \times n$ matrix. Is it true that $\tr ( A^\trans ) = \tr(A)$? If it is true, prove it. If not, give a counterexample.
14. Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors.
(a) Prove that $\mathbf{v}^\trans \mathbf{w} = \mathbf{w}^\trans \mathbf{v}$.
(b) Provide an example to show that $\mathbf{v} \mathbf{w}^\trans$ is not always equal to $\mathbf{w} \mathbf{v}^\trans$.

15. Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors. Prove that $\tr ( \mathbf{v} \mathbf{w}^\trans ) = \mathbf{v}^\trans \mathbf{w}$.
16. Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$.
17. Is it true that a real square matrix $A$ must commute with its transpose $A^{\trans}$?
18. Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix.
19. Calculate the following expressions, using the following matrices:
$A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$ (a) $A B^\trans + \mathbf{v} \mathbf{v}^\trans$.
(b) $A \mathbf{v} – 2 \mathbf{v}$.
(c) $\mathbf{v}^{\trans} B$.
(d) $\mathbf{v}^\trans \mathbf{v} + \mathbf{v}^\trans B A^\trans \mathbf{v}$.

20. Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector. Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$.
21. Let $A$ be a square matrix such that $A^{\trans}A=A$, where $A^{\trans}$ is the transpose matrix of $A$. Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.
22. Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.
(a) The product $AB$ is symmetric if and only if $AB=BA$.
(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.

23. A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.
24. Let $I$ be the $n\times n$ identity matrix, where $n$ is a positive integer. Prove that there are no $n\times n$ matrices $X$ and $Y$ such that $XY-YX=I$.

25. Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$.
(a) Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of $B$.
(b) Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$.
(c) Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$.

26. For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$.
(a) $A=\begin{bmatrix} 4 & 2\\ 2& 1 \end{bmatrix}$.
(b) $A=\begin{bmatrix} 2 & 1\\ 1& 3 \end{bmatrix}$.

27. (a) Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ does not have a square root. Namely, show that there is no complex matrix $B$ such that $B^2=A$.
(b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root matrices.

28. Let $D=\begin{bmatrix} d_1 & 0 & \dots & 0 \\ 0 &d_2 & \dots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \dots & d_n \end{bmatrix}$ be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$. Let $A=(a_{ij})$ be an $n\times n$ matrix such that $A$ commutes with $D$, that is, $AD=DA$. Then prove that $A$ is a diagonal matrix.