Matrices

Exercise2-2

1. For matrices $A = \left(\begin{array}{rr} 2&-3\\ 4&2 \end{array}\right), \ \ \ B = \left(\begin{array}{rr} -1&2\\ 3&0 \end{array}\right )$, evaluate the followings:

(a) $A +B$ (b) $2A - 3B$ (c) $AB, BA$

2. For matrices $A = \left(\begin{array}{rrr} 3&1&7\\ 5&2&-4 \end{array}\right), \ \ \ B = \left(\begin{array}{rr} 2&-3\\ 3&6\\ 4&1 \end{array}\right )$, find $AB, BA$．

3. For the matrix $A = \left(\begin{array}{rrr} 2&3&0\\ 1&4&1\\ 2&0&1 \end{array}\right)$, calculate $A^{2} - 5A + 6I$．

4. Let $A$ and $B$ be symmetric matrices of the order $n$. Show that $A +B$ is a symmetric matrix．

5. Let $A$ and $B$ be symmetric matrices of the order $n$. Find the necessary and sufficient conditions so that $AB$ is a symmetric matrix．

6. Suppose that $A$ is a skew symmetric matrix. Then show that $A^{2}$ is a symmetric marix.

7. Find matrices so that the product of $\left(\begin{array}{ccc} a_{1}&0&0\\ 0&a_{2}&0\\ 0&0&a_{3} \end{array}\right)$ is interchangeable．Here, $a_{1},a_{2},a_{3}$ are different real numbers．

8. Show that any square matrix $A$ can be expressed by the sum of a symmetric matrix and a skew symmetric matrix.

9. Find the product of $A$ and $B$, where $A = \left(\begin{array}{cc\vert c} 1 & 1 & 1\\ 2 & -1 & 0\\ \hline -1 & 0 & ... ... 1 & 2 & 3 & -1\\ 3 & -1 & 1 & 0\\ \hline 0 & 0 & -3 & 1 \end{array}\right)$．