Sets of Linear Equations; Determinants

Exercise 1 : Gaussian and Gauss-Jordan elimination

Solve the following equations using either Gaussian or Gauss-Jordan elimination. Use matrix notation.

$$ \begin{cases} 2x_1 + x_2 + 5x_3 = -21 \\ x_1 + 5x_2 + 2x_3 = 19 \\ 5x_1 + 2x_2 + x_3 = 2 \end{cases} $$
  1. $$ \begin{cases} x - y + 3z = 4 \\ 23x + 2y + 4z = 13 \\ 11.5x + y + 2z = 6.5 \end{cases} $$
  2. $$ \begin{cases} 3x_1 + 2x_2 + x_3 = 49 \\ x_1 + x_2 + x_3 = 8 \\ 5x_1 - 3x_2 + x_3 = 0 \end{cases} $$
  3. $$ \begin{cases} 1.2x - 0.9y + 1.5z = 2.4 \\ 0.8x - 0.5y + 2.5z = 1.8 \\ 1.6x - 1.2y + 2z = 3.2 \end{cases} $$

Exercise 2 : Inverse Matrices

Obtain the inverse of the following matrices:

  1. $$ \begin{pmatrix} 2 & 1 & 0 \\ 1 & 1 & -2 \\ 0 & 3 & -4 \end{pmatrix} $$
  2. $$ \begin{pmatrix} -4 & 8 \\ -6 & 7 \end{pmatrix} $$

Exercise 3 : Homogeneous Equations

Investigate the following sets of homogeneous equations and obtain their solutions.

  1. $$ \begin{cases} x_1 + x_2 - x_3 = 0 \\ -x_1 + 3x_2 + x_3 = 0 \\ x_2 + x_3 = 0 \end{cases} $$
  2. $$ \begin{cases} 2x - 3y + z = 0 \\ 4x + 4y - z = 0 \\ x - \frac{3}{2}y + \frac{1}{2}z = 0 \end{cases} $$

Exercise 4 : Determinants

Evaluate the following determinants:

  1. $$ \begin{vmatrix} 4 & 3 & 2 \\ 1 & 0 & -1 \\ 5 & 2 & 2 \end{vmatrix} $$
  2. $$ \begin{vmatrix} 1 & 7 & 4 & 12 \\ 5 & 5 & 4 & 3 \\ 3 & 4 & 0 & 2 \\ 5 & 35 & 20 & 60 \end{vmatrix} $$
  3. $$ \begin{vmatrix} 6 & 1 & -3 & 1 \\ 0 & 0 & 4 & 0 \\ 4 & 6 & 0 & 7 \\ -3 & 0 & 2 & 8 \end{vmatrix} $$
  4. $$ \begin{vmatrix} 4 & 6 & 0 & 7 \\ -3 & 0 & 2 & 8 \\ 10 & 1 & 0 & 2 \\ 5 & 2 & 0 & 1 \end{vmatrix} $$
  5. $$ \begin{vmatrix} -1 & 4 & 1 & 3 \\ 2 & -2 & -2 & 0 \\ 0 & 0 & -4 & -2 \\ 1 & 0 & 1 & 4 \end{vmatrix} $$

Exercise 5 : Rank of a Matrix

Determine the rank $r$ of the following matrices:

  1. $$ A = \begin{pmatrix} -1 & 4 & 1 & 3 \\ 2 & -2 & -2 & 0 \\ 0 & 0 & -4 & -2 \\ 1 & 0 & 1 & 4 \end{pmatrix} $$
  2. $$ B = \begin{pmatrix} 3 & 2 & 2 & 2 \\ 4 & 2 & 4 & 2 \\ 3 & 1 & 3 & 1 \\ 2 & 1 & 2 & 1 \end{pmatrix} $$