ekiim's blog
    1. Write down the 3 by 3 matrices with entriesaij=ij and bij=ij a_{i j}=i-j \quad \text { and } \quad b_{i j}=\frac{i}{j}
    2. Compute the products ABAB and BABA and A2A^2
  1. For the matrices A=[1021] and B=[1201] 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 ABAB, and BABA and A1A^{-1} and B1B^{-1} and (AB)1(AB)^{-1}

  2. Find examples of 2 by 2 matrices with a12=12a_{12} = \frac12 for with

    1. A2=IA^2 = I
    2. A1=ATA^{-1} = A^T
    3. A2=AA^2 = A
  3. Solve by elimination and back-substituion: u+w=4u+v=3u+v+w=6 and r+w=0u+w=0u+v=6 \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}

  4. Factor the preceding matrices in to A=LUA = LU or PA=LUPA=LU.

    1. There are sixteen 2 by 2 matrices whose entries are 1’s and 0’s How many are invertible?
    2. (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?
  5. There are sixteen matrices 2 by 2 whose entries are 1’s and -1’s. How many are invertible?

  6. How are the rows of EAEA related to the rows of AA if E=[100020401] or E=[111000] or E=[001010100]? 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] ?

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

  8. Find inverses if they exist by inspection or by Gauss-JordanA=[101110011] and A=[210121012] and A=[112121211] 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]

  9. If EE is 2 by 2 and it adds the first equation to the second what are E2E^2 and E8E^8 and 8E8E?

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

    1. If AA is invertible an its rows are in reverse order in BB then BB is invertible.
    2. If AA and BB are summetric then ABAB is symmetric.
    3. If AA and BB are invertible then BABA is invertible
    4. Every nonsingular matrix can be factored in to the product A=LUA = LU of a lower triangular LL and an upper triangular UU.
  11. Solve Ax=bAx = b by solving the triangular systems Lc=bLc=b and Ux=cUx=c.A=LU=[100410101][224013001],b=[001] 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 A1A^{-1} have you found with this particular bb?

  12. If possible find 3 by 3 matrices B such that

    1. BA=2ABA = 2A for every AA
    2. BA=2BBA = 2B for every AA
    3. BABA has the first and last rows of AA reversed
    4. BABA has the first and last columns of AA reversed
  13. Find the value for cc in the following nn by nn inverse:if A=[n111n11111n] then A1=1n+1[c111c11111c] \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]

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

  15. Find the symmetric factorization A=LDLTA = LDL^T of A=[1202640411] and A=[abbc] 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]

  16. Suppose AA is the 4 by 4 identity matrix except for a vector vv in the column 2; A=[1v1000v2000v3100v401] 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]

    1. Factor AA in to LULU assuming v20{v_2} \neq 0
    2. Find A1A^{-1}, which has the same form as AA.
  17. Solve by elimination or show that there is no solution: u+v+w=0u+2v+3w=03u+5u+7w=1 and u+v+w=0u+v+3w=03u+5v+7w=1 \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}

  18. The nn by nn 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 P1(P2P3)=(P1P2)P3P_1(P_2P_3) = (P_1P_2)P_3 is true – because it is true for all matrices.

    1. How may members belong to the group of 4 by 4 matrices and nn by nn permuation matrices?
    2. Find a power k so that all 3 by 3 permutation matrices satisfy Pk=IP^k = I
  19. Describe the rows of DADA and columns of ADAD if D=[2005]D = \begin{bmatrix}2 & 0 \\ 0 & 5\end{bmatrix}.

    1. If AA is invertible wha is the inverse of ATA^T
    2. If A is also symmetric what is the transpose of A1A^{-1}
    3. Illustrate both formulas when A=[2111]A = \begin{bmatrix}2 & 1 \\ 1 & 1\end{bmatrix}.
  20. By experiment with n=2n=2 and n=3n=3 find [2300]n,[2301]n,[2301]1 \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}

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

    1. the parallel plane through the origin
    2. a second plane that also contains the points (6,0,0)(6, 0, 0) and (2,2,0)(2, 2, 0)
    3. a third plane that meets the first and second in the point (4,1,0)(4, 1, 0)
  22. What multiple of row 2 is substracted from row 3 in forward elimination of A=[100210051][120015001] 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 AA is invertible, symmetric and tridiagonal? What are the pivots?

    1. What vector xx will make Ax=Ax= column 1 + 2(column 3), for all 3 by 3 matrices AA?
    2. 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.
  23. True or false, with reason if true and counter example if false:

    1. If L1U1=L2U2L_1 U_1 = L_2 U_2 (upper triangulars UU’s with nonzero diagonal, lower triangular LL’s with unit diagonal) then L1=L2L_1=L_2 and U1=U2U_1=U_2. The LULU factorization is unique.
    2. If A2+A=IA^2 + A = I then A1=A+IA^{-1} = A + I
    3. If all diagonal entries of AA are zero then AA is singular.
  24. By experiment or Gauss-Jordan Method compute [100l10m01]n,[100l10m01]1,[100l100m1]1\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}

  25. Write down the 2 by 2 matrices which

    1. reverse the direction for every vector
    2. project every vector onto the x2x_2-axis
    3. turn every vector counterclowise through 9090^\circ
    4. reflect every vector through the 4545^\circ line x1=x2x_1 = x_2