Precalculus by Richard Wright

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I am the good shepherd. The good shepherd lays down his life for the sheep. John 10:11 NIV

9-Review

Take this test as you would take a test in class. When you are finished, check your work against the answers. On this assignment round your answers to three decimal places unless otherwise directed.

  1. Perform the indicated row operations on \(\left[\begin{matrix} 2 & 4 & -6 \\ -8 & 1 & 2 \\ 0 & 3 & -4 \end{matrix}\right]\)
    1. Add four times the 1st row to the 2nd row.
    2. Multiply the 1st row by \(\frac{1}{2}\).
  2. Solve the system with Gaussian Elimination \(\left\{\begin{alignat}{4} x &-& 3y &+& 2z &=& 5 \\ -2x &+& y && &=& -4 \\ && 2y &-& z &=& -3 \end{alignat}\right.\)
  3. Is the matrix in reduced-row echelon form? \(\left[\begin{matrix} 1 & 0 & 0 & 2 \\ 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 1 \end{matrix}\right]\)
  4. Put the matrix into reduced-row echelon form. \(\left[\begin{matrix} 1 & -2 & 5 & 2 \\ 1 & 3 & 1 & 5 \\ 2 & -1 & 0 & 1 \end{matrix}\right]\)
  5. Perform the indicated operations.
  6. \(\left[\begin{matrix} 2 & -1 & 3 \\ 0 & 7 & -3 \end{matrix}\right] - \left[\begin{matrix} 4 & 0 & -4 \\ 2 & -2 & 3 \end{matrix}\right]\)
  7. \(\left[\begin{matrix} 1 & 3 \\ -2 & 5 \end{matrix}\right] + 2\left[\begin{matrix} -1 & 2 \\ 3 & 1 \end{matrix}\right]\)
  8. \(\left[\begin{matrix} 1 & 2 & -1 & -2 \\ 0 & 3 & 1 & -2 \end{matrix}\right] \left[\begin{matrix} 3 & 0 \\ -1 & -2 \\ 2 & 5 \\ 1 & 1 \end{matrix}\right]\)
  9. Find the inverse of \(\left[\begin{matrix} 2 & -1 \\ -2 & 3 \end{matrix}\right]\)
  10. Find the inverse of \(\left[\begin{matrix} 1 & 0 & 1 \\ -1 & 2 & 4 \\ 2 & 1 & -1 \end{matrix}\right]\)
  11. Use an inverse matrix to solve \(\left\{\begin{align} x + z &= 5 \\ -x + 2y + 4z &= 11 \\ 2x + y - z &= -4 \end{align}\right.\)
  12. Find \(\left|\begin{matrix} 2 & -1 \\ 4 & -3 \end{matrix}\right|\)
  13. Find \(\left|\begin{matrix} 1 & -2 & -1 \\ 0 & -3 & 2 \\ 0 & 2 & 2 \end{matrix}\right|\) using the shortcut.
  14. Find \(\left|\begin{matrix} 3 & 1 & -2 \\ 4 & 2 & 0 \\ 1 & -2 & -1 \end{matrix}\right|\) using the expansion by cofactors.
  15. Use Cramer's Rule to solve \(\left\{\begin{align} 2x + 3y - z &= 0 \\ y + z &= 0 \\ -x + 2y - z &= -10 \end{align}\right.\)
  16. Find the area of triangle with vertices (−3, 2), (2, −1), (3, 5).
  17. Use a determinant to find the equation of the line through (3, 1) and (−2, 4).
  18. Use \(\left[\begin{matrix} 2 & 3 \\ -1 & 1 \end{matrix}\right]\) to encode the message I GOT A.

Answers

    1. \(\left[\begin{matrix} 2 & 4 & -6 \\ 0 & 17 & -22 \\ 0 & 3 & -4 \end{matrix}\right]\)
    2. \(\left[\begin{matrix} 1 & 2 & -3 \\ -8 & 1 & 2 \\ 0 & 3 & -4 \end{matrix}\right]\)
  2. (1, −2, −1)
  3. No, the 2 and 5 should be zeros.
  4. \(\left[\begin{matrix} 1 & 0 & 0 & \frac{20}{19} \\ 0 & 1 & 0 & \frac{21}{19} \\ 0 & 0 & 1 & \frac{12}{19} \end{matrix}\right]\)
  5. \(\left[\begin{matrix} -2 & -1 & 7 \\ -2 & 9 & -6 \end{matrix}\right]\)
  6. \(\left[\begin{matrix} -1 & 7 \\ 4 & 7 \end{matrix}\right]\)
  7. \(\left[\begin{matrix} -3 & -11 \\ -3 & -3 \end{matrix}\right]\)
  8. \(\left[\begin{matrix} \frac{3}{4} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{2} \end{matrix}\right]\)
  9. \(\left[\begin{matrix} \frac{6}{11} & -\frac{1}{11} & \frac{2}{11} \\ -\frac{7}{11} & \frac{3}{11} & \frac{5}{11} \\ \frac{5}{11} & \frac{1}{11} & -\frac{2}{11} \end{matrix}\right]\)
  10. (1, −2, 4)
  11. −2
  12. −10
  13. 18
  14. (4, −2, 2)
  15. \(\frac{33}{2}\)
  16. 3x + 5y = 14
  17. 18, 27, −1, 36, 40, 60, 2, 3