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. Prove the following identity for any positive integer $n$.
    \[\begin{bmatrix}
    \cos \theta & -\sin \theta\\
    \sin \theta& \cos \theta
    \end{bmatrix}^n=\begin{bmatrix}
    \cos n\theta & -\sin n\theta\\
    \sin n\theta& \cos n\theta
    \end{bmatrix}.\]
  11. Is it true that a real square matrix $A$ must commute with its transpose $A^{\trans}$?
  12. 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.
  13. 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 $AB=BA$.

  14. A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.
  15. 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
    \[XY-YX=I.\]
  16. 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 $\tr(A^2)>0$.

  17. 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) $A=\begin{bmatrix}
    2 & 1\\
    1& 3
    \end{bmatrix}$.

  18. (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.

  19. 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.