- Write down the 3 by 3 matrices with entries$a_{i j}=i-j \quad \text { and } \quad b_{i j}=\frac{i}{j}$
- Compute the products $AB$ and $BA$ and $A^2$

For the matrices $A=\left[ \begin{array}{ll}{1} & {0} \\ {2} & {1}\end{array}\right] \quad \text { and } \quad B=\left[ \begin{array}{ll}{1} & {2} \\ {0} & {1}\end{array}\right]$ compute $AB$, and $BA$ and $A^{-1}$ and $B^{-1}$ and $(AB)^{-1}$

Find examples of 2 by 2 matrices with $a_{12} = \frac12$ for with

- $A^2 = I$
- $A^{-1} = A^T$
- $A^2 = A$

Solve by elimination and back-substituion: $\begin{aligned} u +w&=4 \\ u+v &=3 \\ u+v+w &=6 \end{aligned} \text { and } \begin{aligned} r+w &=0 \\ u &+w=0 \\ u+v &=6 \end{aligned}$

Factor the preceding matrices in to $A = LU$ or $PA=LU$.

- There are sixteen 2 by 2 matrices whose entries are 1’s and 0’s How many are invertible?
- (Much harder!) If you put 1’s and 0’s at random into the entires of a 10 by 10 matrix its more likely to be invertible or singular?

There are sixteen matrices 2 by 2 whose entries are 1’s and -1’s. How many are invertible?

How are the rows of $EA$ related to the rows of $A$ if $E=\left[ \begin{array}{lll}{1} & {0} & {0} \\ {0} & {2} & {0} \\ {4} & {0} & {1}\end{array}\right] \quad \text { or } \quad E=\left[ \begin{array}{lll}{1} & {1} & {1} \\ {0} & {0} & {0}\end{array}\right] \text { or } E=\left[ \begin{array}{lll}{0} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {0}\end{array}\right] ?$

Write down a 2 by 2 system with infinitely many solutions.

Find inverses if they exist by inspection or by Gauss-Jordan$A=\left[ \begin{array}{lll}{1} & {0} & {1} \\ {1} & {1} & {0} \\ {0} & {1} & {1}\end{array}\right] \text { and } A=\left[ \begin{array}{ccc}{2} & {1} & {0} \\ {1} & {2} & {1} \\ {0} & {1} & {2}\end{array}\right] \text { and } A=\left[ \begin{array}{rrr}{1} & {1} & {-2} \\ {1} & {-2} & {1} \\ {-2} & {1} & {1}\end{array}\right]$

If $E$ is 2 by 2 and it adds the first equation to the second what are $E^2$ and $E^8$ and $8E$?

True or fals, with reason if true or counterexample if false:

- If $A$ is invertible an its rows are in reverse order in $B$ then $B$ is invertible.
- If $A$ and $B$ are summetric then $AB$ is symmetric.
- If $A$ and $B$ are invertible then $BA$ is invertible
- Every nonsingular matrix can be factored in to the product $A = LU$ of a lower triangular $L$ and an upper triangular $U$.

Solve $Ax = b$ by solving the triangular systems $Lc=b$ and $Ux=c$.$A=L U=\left[ \begin{array}{lll}{1} & {0} & {0} \\ {4} & {1} & {0} \\ {1} & {0} & {1}\end{array}\right] \left[ \begin{array}{lll}{2} & {2} & {4} \\ {0} & {1} & {3} \\ {0} & {0} & {1}\end{array}\right], \quad b=\left[ \begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right]$ What part of $A^{-1}$ have you found with this particular $b$?

If possible find 3 by 3 matrices B such that

- $BA = 2A$ for every $A$
- $BA = 2B$ for every $A$
- $BA$ has the first and last rows of $A$ reversed
- $BA$ has the first and last columns of $A$ reversed

Find the value for $c$ in the following $n$ by $n$ inverse:$\text{if } \quad A=\left[ \begin{array}{rrrr}{n} & {-1} & {\cdot} & {-1} \\ {-1} & {n} & {\cdots} & {-1} \\ {\cdot} & {\cdot} & {\cdot} & {-1} \\ {-1} & {-1} & {-1} & {n}\end{array}\right] \text { then } A^{1}=\frac{1}{n+1} \left[ \begin{array}{cccc}{c} & {1} & {\cdot} & {1} \\ {1} & {c} & {\cdot} & {1} \\ {\cdot} & {\cdot} & {\cdot} & {1} \\ {1} & {1} & {1} & {c}\end{array}\right]$

for which values of $k$ does $\begin{aligned} k x+y &=1 \\ x+k y &=1 \end{aligned}$ have no solution, one solution or infinitely many solutions?

Find the symmetric factorization $A = LDL^T$ of $A=\left[ \begin{array}{lll}{1} & {2} & {0} \\ {2} & {6} & {4} \\ {0} & {4} & {11}\end{array}\right] \quad \text { and } \quad A=\left[ \begin{array}{ll}{a} & {b} \\ {b} & {c}\end{array}\right]$

Suppose $A$ is the 4 by 4 identity matrix except for a vector $v$ in the column 2; $A=\left[ \begin{array}{llll}{1} & {v_{1}} & {0} & {0} \\ {0} & {v_{2}} & {0} & {0} \\ {0} & {v_{3}} & {1} & {0} \\ {0} & {v_{4}} & {0} & {1}\end{array}\right]$

- Factor $A$ in to $LU$ assuming ${v_2} \neq 0$
- Find $A^{-1}$, which has the same form as $A$.

Solve by elimination or show that there is no solution: $\begin{array}{r}{u+v+w=0} \\ {u+2 v+3 w=0} \\ {3 u+5 u+7 w=1}\end{array} \text { and } \begin{aligned} u+v+w &=0 \\ u+v+3 w &=0 \\ 3 u+5 v+7 w &=1 \end{aligned}$

The $n$ by $n$ permuation matrices are an important example of a

*“group”*. If you multiply them you stay inside the group; they have inverses in the group; the identity is in the group; and the laws $P_1(P_2P_3) = (P_1P_2)P_3$ is true – because it is true for all matrices.- How may members belong to the group of 4 by 4 matrices and $n$ by $n$ permuation matrices?
- Find a power k so that all 3 by 3 permutation matrices satisfy $P^k = I$

Describe the rows of $DA$ and columns of $AD$ if $D = \begin{bmatrix}2 & 0 \\ 0 & 5\end{bmatrix}$.

- If $A$ is invertible wha is the inverse of $A^T$
- If A is also symmetric what is the transpose of $A^{-1}$
- Illustrate both formulas when $A = \begin{bmatrix}2 & 1 \\ 1 & 1\end{bmatrix}$.

By experiment with $n=2$ and $n=3$ find $\left[ \begin{array}{ll}{2} & {3} \\ {0} & {0}\end{array}\right]^{n}, \left[ \begin{array}{ll}{2} & {3} \\ {0} & {1}\end{array}\right]^{n}, \left[ \begin{array}{ll}{2} & {3} \\ {0} & {1}\end{array}\right]^{-1}$

Starting with a first plane $u + 2v -w = 6$ find the equations for

- the parallel plane through the origin
- a second plane that also contains the points $(6, 0, 0)$ and $(2, 2, 0)$
- a third plane that meets the first and second in the point $(4, 1, 0)$

What multiple of row 2 is substracted from row 3 in forward elimination of $A=\left[ \begin{array}{lll}{1} & {0} & {0} \\ {2} & {1} & {0} \\ {0} & {5} & {1}\end{array}\right] \left[ \begin{array}{lll}{1} & {2} & {0} \\ {0} & {1} & {5} \\ {0} & {0} & {1}\end{array}\right]$ How do you know (without multiplying those factors) that $A$ is invertible, symmetric and tridiagonal? What are the pivots?

- What vector $x$ will make $Ax=$ column 1 + 2(column 3), for all 3 by 3 matrices $A$?
- Construct a matrix which has column 1 + 2(column 3) = 0. Check that A is singular (fewer than 3 pivots) and explain why that must happen.

True or false, with reason if true and counter example if false:

- If $L_1 U_1 = L_2 U_2$ (upper triangulars $U$’s with nonzero diagonal, lower triangular $L$’s with unit diagonal) then $L_1=L_2$ and $U_1=U_2$. The $LU$ factorization is unique.
- If $A^2 + A = I$ then $A^{-1} = A + I$
- If all diagonal entries of $A$ are zero then $A$ is singular.

By experiment or Gauss-Jordan Method compute $\left[ \begin{array}{lll}{1} & {0} & {0} \\ {l} & {1} & {0} \\ {m} & {0} & {1}\end{array}\right]^{n}, \quad \left[ \begin{array}{lll}{1} & {0} & {0} \\ {l} & {1} & {0} \\ {m} & {0} & {1}\end{array}\right]^{-1}, \quad \left[ \begin{array}{ccc}{1} & {0} & {0} \\ {l} & {1} & {0} \\ {0} & {m} & {1}\end{array}\right]^{-1}$

Write down the 2 by 2 matrices which

- reverse the direction for every vector
- project every vector onto the $x_2$-axis
- turn every vector counterclowise through $90^\circ$
- reflect every vector through the $45^\circ$ line $x_1 = x_2$