ekiim's blog
1. When is an upper triangular matrix non singular?

2. What multiple of row 2 of $A$ will elimination subtract from row 3?$A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 1 & 4 & 1 \end{bmatrix} \begin{bmatrix} 5 & 7 & 8 \\ 0 & 2 & 3 \\ 0 & 0 & 6 \end{bmatrix}$ What will be the pivots? Will a row exchange be required?

3. Multiply the matrix $L = E^{-1}F^{-1}G^{-1}$ in equation (6) by $GFE$ in equation (3): $\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ -1 & -1 & 1 \end{bmatrix} \text{ times } \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ -1 & 1 & 1 \end{bmatrix}$ Multiply also in the opposite order. Why are the answers what they are?

4. Apply elimination to produce the factors L and U for $A=\left[ \begin{array}{ll}{2} & {1} \\ {8} & {7}\end{array}\right] \text { and } A=\left[ \begin{array}{lll}{3} & {1} & {1} \\ {1} & {3} & {1} \\ {1} & {1} & {3}\end{array}\right] \quad \text { and } \quad A=\left[ \begin{array}{lll}{1} & {1} & {1} \\ {1} & {4} & {4} \\ {1} & {4} & {8}\end{array}\right]$

5. Factor $A$ in to $LU$ and write down the upper triangular system $Ux = c$ which appears afeter elimination for$A x=\left[ \begin{array}{lll}{2} & {3} & {3} \\ {0} & {5} & {7} \\ {6} & {9} & {8}\end{array}\right] \left[ \begin{array}{l}{u} \\ {v} \\ {w}\end{array}\right]=\left[ \begin{array}{l}{2} \\ {2} \\ {5}\end{array}\right]$

6. Find $E^2$ and $E^8$ and $E^{-1}$ if $E=\left[ \begin{array}{ll}{1} & {0} \\ {6} & {1}\end{array}\right]$

7. Find the products $FGH$ and $HGF$ if (with upper triangular zeros omitted) $F=\left[ \begin{array}{lll}{1} & & \\ {2} & {1} & \\ {0} & {0} & {1} \\ {0} & {0} & {0} & {1}\end{array}\right], G=\left[ \begin{array}{lll}{1} & & \\ {0} & {1} & & \\ {0} & {2} & {1} \\ {0} & {0} & {0} & {1}\end{array}\right] H=\left[ \begin{array}{cccc}{1} & & & \\ {0} & {1} & & \\ {0} & {0} & {1} & \\ {0} & {0} & {2} & {1}\end{array}\right]$

8. (Second proof of $A = LU$) The third row of $U$ comes from the third row of $A$ by subtracting multiples of rows 1 and 2 (of $U$!):$\operatorname{row} 3 \text { of } U=\operatorname{row} 3 \text { of } A-l_{31}(\text { row } 1 \text { of } U)-I_{32}(\text { row } 2 \text { of } U)$

1. Why are rows of $U$ substractd off and not rows of $A$, Answer: Because by the time a pivot row is used…
2. The equation above is the same as $\text{row } 3 \text { of } A=l_{31}(\text { row } 1 \text { of } U)+l_{32}(\text { row } 2 \text { of } U\}+1(\text { row } 3 \text { of } U)$ Which rule for matrix multiplation (shaped in 1D) makes this exactly row 3 of $L$ times $U$? The other rows of $LU$ agree similarly with the rows of $A$.
1. Under what conditions is $A$ nonsingular, if $A$ is the product $A=\left[ \begin{array}{rrr} {1} & {0} & {0} \\ {-1} & {1} & {0} \\ {0} & {-1} & {1}\end{array}\right] \left[ \begin{array}{ccc}{d_1} & & \\ & {d_{2}} & \\ & & d_3\end{array}\right] \left[ \begin{array}{rrr}{1} & {-1} & {0} \\ {0} & {1} & {-1} \\ {0} & {0} & {1}\end{array}\right]\text{ ?}$
2. Solve the system $Ax = b$ with $Lc=b$:$\left[ \begin{array}{rrr}{1} & {0} & {0} \\ {-1} & {1} & {0} \\ {0} & {-1} & {1}\end{array}\right] \left[ \begin{array}{l}{c_{1}} \\ {c_{2}} \\ {c_{3}}\end{array}\right]=\left[ \begin{array}{l}{0} \\ {0} \\ {1}\end{array}\right]=b$
1. Why does it take approximately $n^2/2$ multiplication-substraction setps to solve each of $Lc=b$ and $Ux=c$?
2. How many steps does elimination use in solving 10 sysmtes with the same 60 by 60 coefficient matrix $A$?
9. Solve $L U x=\left[ \begin{array}{lll}{1} & {0} & {0} \\ {1} & {1} & {0} \\ {1} & {0} & {1}\end{array}\right] \left[ \begin{array}{lll}{2} & {4} & {4} \\ {0} & {1} & {2} \\ {0} & {0} & {1}\end{array}\right] \left[ \begin{array}{l}{u} \\ {v} \\ {w}\end{array}\right]=\left[ \begin{array}{l}{2} \\ {0} \\ {2}\end{array}\right]$, without multiplying $LU$ to find $A$

10. How could you factor A into a product $UL$, upper triangular times lower triangular? Would they be the same factors as $A = LU$?

11. Solve by elimination, exchanging rows when necessary:\begin{aligned} u+4 v+2 w &=-2 \\-2 u-8 v+3 w &=32 \\ v+w &=1 \end{aligned} \text { and } \begin{aligned} v+w &=0 \\ u+v &=0 \\ u+v+w &=1 \end{aligned} Which permiutation matrices are required?

12. Write down all six of the 3 by 3 permuation matrices, including $P = 1$. Idenfity inverses, which are also permuation matrices — they satisfy $PP^{-1}=I$ and are on the same list.

13. Find the $PA = LDU$ factorization (and check them) for $A=\left[ \begin{array}{lll}{0} & {1} & {1} \\ {1} & {0} & {1} \\ {2} & {3} & {4}\end{array}\right] \quad \text { and } \quad A=\left[ \begin{array}{ccc}{1} & {2} & {1} \\ {2} & {4} & {2} \\ {1} & {1} & {1}\end{array}\right]$

14. Find a nonsingular 4 by 4 matrix that requires three rows exchanges to reach the end of elimination. If possible let the example be a permuation matrix.

15. In the $LPU$ order that albegrists prefere, elimination exchanges rows only at the end: $A=\left[ \begin{array}{lll}{1} & {1} & {1} \\ {1} & {1} & {3} \\ {2} & {5} & {8}\end{array}\right] \rightarrow \left[ \begin{array}{lll}{1} & {1} & {1} \\ {0} & {0} & {2} \\ {0} & {3} & {6}\end{array}\right]=P U=\left[ \begin{array}{lll}{1} & {0} & {0} \\ {0} & {0} & {1} \\ {0} & {1} & {0}\end{array}\right] \left[ \begin{array}{lll}{1} & {1} & {1} \\ {0} & {3} & {6} \\ {0} & {0} & {2}\end{array}\right]$ What is the $L$ in this case? Unlike $PA = LU$ and the examples after $1J$ the multiplers stay in place ($l_{21}$ is 1 and $l_{31}$ is 2).

16. Decide whether the following systems are singluar or nonsingular, and whether they have no solution, one solution, or infinitely many solutions:\begin{array}{r}{v-w=2} \\ {u-v \quad=2} \\ {u \quad-w=2}\end{array} \text { and } \begin{aligned} v-w &=0 \\ u-v &=0 \\ u &=0 \end{aligned} \text { and } \begin{array}{r}{v+w=1} \\ {u+v \quad=1} \\ {u \quad+w=1}\end{array}

17. Which values of $a, b, c$, lead to row exchanges, and which make the matrices singular?$A=\left[ \begin{array}{lll}{1} & {2} & {0} \\ {a} & {8} & {3} \\ {0} & {b} & {5}\end{array}\right] \quad \text { and } \quad A=\left[ \begin{array}{ll}{c} & {2} \\ {6} & {4}\end{array}\right]$