Practice questions

Various practice questions will be posted here as the semester progresses. These questions are from previous exams, assignments and tutorials. No solutions will be posted.

Tutorial 3, 2009

• Convert into decimal representation: (a) \((10011101)_2\); (b) \((0.001101)_2\)
• Convert into binary representation: (a) \((45.625)_{10}\); (b) \((0.1)_{10}\)
• Consider the following system of linear algebraic equations.
  • Use Gauss elimination (without pivoting) to solve these equations for \((x_1,x_2,x_3)\).
  • Validate your solution by comparing it to the one obtained with computer software.

\[ \begin{align*} \left\{\begin{array}{rcl} 2 x_1 -2x_2 +4 x_3 & = & 0 \\ x_1 -3 x_2 + 4x_3 & = & -1 \\ 3x_1 - x_2 +5x_3 &= & 0 \end{array}\right. \end{align*} \]

Tutorial 4, 2009

• We have already seen several elimination techniques for linear equation solving. In this question, you are to solve the following linear algebraic equations, using (a) Gauss elimination, (b) LU decomposition

\[\begin{align*} \left\{\begin{array}{rcl} 2 x_1 -2x_2 +4 x_3 & = & 0 \\ x_1 -3 x_2 + 4x_3 & = & -1 \\ 3x_1 - x_2 +5x_3 &= & 0 \end{array}\right. \end{align*} \]

• We have also seen that the LU decomposition technique can be used to calculate matrix inverses. Using the results obtained in the question above, compute the inverse of the following matrix,

\[ \begin{align*} A &= \left[ \begin{array}{rrr} 2 & -2 & 4 \\ 1 & -3 & 4 \\ 3 & -1 & 5 \end{array} \right] \end{align*} \]