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Elimination

Row operations carrying an augmented matrix to reduced row echelon form, free variables, consistency, rank, and inverses computed by the same procedure.

Notes on Diffy Qs: Differential Equations for Engineers, Elimination

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8 exercises

Exercise 101

Compute the reduced row echelon form for the following matrices:

  1. [101010]\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}
  2. [1234]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
  3. [11−2−2]\begin{bmatrix} 1 & 1 \\ -2 & -2 \end{bmatrix}
  4. [1−3146−2−26−2]\begin{bmatrix} 1 & -3 & 1 \\ 4 & 6 & -2 \\ -2 & 6 & -2 \end{bmatrix}
  5. [22521−24−1031−2]\begin{bmatrix} 2 & 2 & 5 & 2 \\ 1 & -2 & 4 & -1 \\ 0 & 3 & 1 & -2 \end{bmatrix}
  6. [−264360−3042−11]\begin{bmatrix} -2 & 6 & 4 & 3 \\ 6 & 0 & -3 & 0 \\ 4 & 2 & -1 & 1 \end{bmatrix}
  7. [00000000]\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}
  8. [12331235]\begin{bmatrix} 1 & 2 & 3 & 3 \\ 1 & 2 & 3 & 5 \end{bmatrix}
2.5

Exercise 102

Compute the inverse of the given matrices

  1. [010−100001]\begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}
  2. [111110100]\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}
  3. [240223241]\begin{bmatrix} 2 & 4 & 0 \\ 2 & 2 & 3 \\ 2 & 4 & 1 \end{bmatrix}
2.5

Exercise 103

Solve (find all solutions), or show no solution exists

  1. 4x1+3x2=−15x1+6x2=4\begin{aligned} 4x_1+3x_2 & = -1 \\ 5x_1+6x_2 & = 4 \end{aligned}
  2. 5x+6y+5z=76x+8y+6z=−15x+2y+5z=2\begin{aligned} 5x+6y+5z & = 7 \\ 6x+8y+6z & = -1 \\ 5x+2y+5z & = 2 \end{aligned}
  3. a+5b+6c=−1a+5b+6c=−1−2a+5b+6c=8\begin{aligned} a+\phantom{5}b+\phantom{6}c & = -1 \\ a+5b+6c & = -1 \\ -2a+5b+6c & = 8 \end{aligned}
  4. −2x1+2x2+8x3=6x2+8x3=2x1+4x2+8x3=7\begin{aligned} -2 x_1+2x_2+8x_3 & = 6 \\ x_2+\phantom{8}x_3 & = 2 \\ x_1+4x_2+\phantom{8}x_3 & = 7 \end{aligned}
2.5

Exercise 104

By computing the inverse, solve the following systems for x⃗\vec{x}.

  1. [−1133]x⃗=[46]\begin{bmatrix} -1 & 1 \\ 3 & 3 \end{bmatrix} \vec{x} = \begin{bmatrix} 4 \\ 6 \end{bmatrix}
  2. [2716]x⃗=[13]\begin{bmatrix} 2 & 7 \\ 1 & 6 \end{bmatrix} \vec{x} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}
2.5

Exercise 105

Compute the rank of the given matrices

  1. [7−16777762]\begin{bmatrix} 7 & -1 & 6 \\ 7 & 7 & 7 \\ 7 & 6 & 2 \end{bmatrix}
  2. [111111222]\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \end{bmatrix}
  3. [03−163147−1]\begin{bmatrix} 0 & 3 & -1 \\ 6 & 3 & 1 \\ 4 & 7 & -1 \end{bmatrix}
2.5

Exercise 106

For the matrices in \exerciseref{exercise:rankmatrixans}, find a linearly independent set of row vectors that span the row space (they do not need to be rows of the matrix).

2.5

Exercise 107

For the matrices in \exerciseref{exercise:rankmatrixans}, find a linearly independent set of columns that span the column space. That is, find the pivot columns of the matrices.

2.5

Exercise 108

Find a linearly independent subset of the following vectors that has the same span.

2.5

Adapted from Notes on Diffy Qs © 2008–2026 Jiří Lebl. Licensed under CC BY-SA 4.0. Source: Notes on Diffy Qs: Differential Equations for Engineers. Imported at revision 658bcae.