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

  2. What multiple of row 2 of AA will elimination subtract from row 3?A=[100210141][578023006] 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=E1F1G1L = E^{-1}F^{-1}G^{-1} in equation (6) by GFEGFE in equation (3): [100210111] times [100210111] \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=[2187] and A=[311131113] and A=[111144148] 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 AA in to LULU and write down the upper triangular system Ux=cUx = c which appears afeter elimination forAx=[233057698][uvw]=[225] 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 E2E^2 and E8E^8 and E1E^{-1} if E=[1061]E=\left[ \begin{array}{ll}{1} & {0} \\ {6} & {1}\end{array}\right]

  7. Find the products FGHFGH and HGFHGF if (with upper triangular zeros omitted) F=[1210010001],G=[1010210001]H=[1010010021] 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=LUA = LU) The third row of UU comes from the third row of AA by subtracting multiples of rows 1 and 2 (of UU!):row3 of U=row3 of Al31( row 1 of U)I32( row 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 UU substractd off and not rows of AA, Answer: Because by the time a pivot row is used…
    2. The equation above is the same as row 3 of A=l31( row 1 of U)+l32( row 2 of U}+1( row 3 of U) \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 LL times UU? The other rows of LULU agree similarly with the rows of AA.
    1. Under what conditions is AA nonsingular, if AA is the product A=[100110011][d1d2d3][110011001] ?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=bAx = b with Lc=bLc=b:[100110011][c1c2c3]=[001]=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 n2/2n^2/2 multiplication-substraction setps to solve each of Lc=bLc=b and Ux=cUx=c?
    2. How many steps does elimination use in solving 10 sysmtes with the same 60 by 60 coefficient matrix AA?
  9. Solve LUx=[100110101][244012001][uvw]=[202]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 LULU to find AA

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

  11. Solve by elimination, exchanging rows when necessary:u+4v+2w=22u8v+3w=32v+w=1 and v+w=0u+v=0u+v+w=1 \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=1P = 1. Idenfity inverses, which are also permuation matrices — they satisfy PP1=IPP^{-1}=I and are on the same list.

  13. Find the PA=LDUPA = LDU factorization (and check them) for A=[011101234] and A=[121242111] 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 LPULPU order that albegrists prefere, elimination exchanges rows only at the end: A=[111113258][111002036]=PU=[100001010][111036002] 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 LL in this case? Unlike PA=LUPA = LU and the examples after 1J1J the multiplers stay in place (l21l_{21} is 1 and l31l_{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:vw=2uv=2uw=2 and vw=0uv=0u=0 and v+w=1u+v=1u+w=1\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,ca, b, c, lead to row exchanges, and which make the matrices singular?A=[120a830b5] and A=[c264] 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]