Matrix multiplication

Let $A=\left[\begin{array}{rrr}1&2&0\\ -1&1&1\end{array}\right]$. Multiply $A$ by each of the following vectors.

Compute the following by columns and by rows. Convince yourself that both method give the same result.

Write the vector

in the form $A\left[\begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{array}\right]$ where $A$ is an appropriate matrix.

Write the vector

in the form $A\left[\begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\end{array}\right]$ where $A$ is an appropriate matrix.

Consider the matrices

Find the following if possible. If it is not possible explain why.

1. $-3A$

2. $3B-A$

3. $AC$

4. $CB$

5. $AE$

6. $EA$

Consider the matrices

Find the following if possible. If it is not possible explain why.

1. $-3A$

2. $3B-A$

3. $AC$

4. $CA$

5. $AE$

6. $EA$

7. $BE$

8. $DE$

Let $A=\left[\begin{array}{rr}-1&-1\\ 3&3\end{array}\right]$. Find all $2\times 2$-matrices $B$ such that $AB=0$.

Let $A=\left[\begin{array}{rr}1&2\\ 3&4\end{array}\right]$ and $B=\left[\begin{array}{rr}1&2\\ 3&k\end{array}\right]$. Is it possible to find $k$ such that $AB=BA$? If so, what should $k$ equal?

Let $A=\left[\begin{array}{rr}1&2\\ 3&4\end{array}\right]$ and $B=\left[\begin{array}{rr}1&2\\ 1&k\end{array}\right]$. Is it possible to choose $k$ such that $AB=BA$? If so, what should $k$ equal?

Compute $A^{4}$, where

Hint: you can save some work by calculating $A^{2}$ times $A^{2}$.

Find $2\times 2$-matrices $A$, $B$, and $C$ such that $A\neq 0$, $C\neq B$, but $AC=AB$.

Find $2\times 2$-matrices $A$ and $B$ such that $A\neq 0$ and $B\neq 0$ but $AB=0$.

Find $3\times 3$-matrices $A$ and $B$ such that $AB\neq BA$.

Give an example of a matrix $A$ such that $A^{2}=I$ and yet $A\neq I$ and $A\neq-I$.

Suppose $A$ and $B$ are square matrices of the same size. Which of the following are necessarily true?

1. $(A+B)^{2}=A^{2}+2AB+B^{2}$.

2. $(A-B)^{2}=A^{2}-2AB+B^{2}$.

3. $(AB)^{2}=A^{2}B^{2}$.

4. $(A+B)^{2}=A^{2}+AB+BA+B^{2}$.

5. $A^{2}B^{2}=A(AB)B$.

6. $(A+B)^{3}=A^{3}+3A^{2}B+3AB^{2}+B^{3}$.

7. $(A+B)(A-B)=A^{2}-B^{2}$.