Row Reduction — Practice Problems and Worked Solutions

17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the reduced row echelon form.

   $$A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\end{array}\right]$$

   Answer: $\left[\begin{array}{ccc}1& 0& -5\\ 0& 1& 4\end{array}\right]$

   Step-by-step solution

   1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& -5\\ 0& 1& 4\end{array}\right]$$
   3. $$\text{Pivot columns: }\left(0,1\right)$$
2. 2.

   Find the reduced row echelon form.

   $$A=\left[\begin{array}{ccc}1& 1& 2\\ 2& 2& 4\end{array}\right]$$

   Answer: $\left[\begin{array}{ccc}1& 1& 2\\ 0& 0& 0\end{array}\right]$

   Step-by-step solution

   1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 1& 2\\ 0& 0& 0\end{array}\right]$$
   3. $$\text{Pivot columns: }\left(0,\right)$$
3. 3.

   Find the reduced row echelon form.

   $$A=\left[\begin{array}{ccc}1& -1& 0\\ 2& 1& 3\end{array}\right]$$

   Answer: $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\end{array}\right]$

   Step-by-step solution

   1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\end{array}\right]$$
   3. $$\text{Pivot columns: }\left(0,1\right)$$
4. 4.

   Find the reduced row echelon form.

   $$A=\left[\begin{array}{ccc}2& 4& 2\\ 1& 3& 1\end{array}\right]$$

   Answer: $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\end{array}\right]$

   Step-by-step solution

   1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\end{array}\right]$$
   3. $$\text{Pivot columns: }\left(0,1\right)$$
5. 5.

   Find the reduced row echelon form.

   $$A=\left[\begin{array}{ccc}1& 2& -1\\ 3& 1& 4\end{array}\right]$$

   Answer: $\left[\begin{array}{ccc}1& 0& \frac{9}{5}\\ 0& 1& -\frac{7}{5}\end{array}\right]$

   Step-by-step solution

   1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& \frac{9}{5}\\ 0& 1& -\frac{7}{5}\end{array}\right]$$
   3. $$\text{Pivot columns: }\left(0,1\right)$$
6. 6.

   Find the reduced row echelon form.

   $$A=\left[\begin{array}{ccc}1& 0& 2\\ 2& 1& 5\\ 3& 1& 7\end{array}\right]$$

   Answer: $\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 1\\ 0& 0& 0\end{array}\right]$

   Step-by-step solution

   1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 1\\ 0& 0& 0\end{array}\right]$$
   3. $$\text{Pivot columns: }\left(0,1\right)$$
7. 7.

   Find the reduced row echelon form.

   $$A=\left[\begin{array}{ccc}1& 3& 2\\ 0& 1& 5\end{array}\right]$$

   Answer: $\left[\begin{array}{ccc}1& 0& -13\\ 0& 1& 5\end{array}\right]$

   Step-by-step solution

   1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& -13\\ 0& 1& 5\end{array}\right]$$
   3. $$\text{Pivot columns: }\left(0,1\right)$$
8. 8.

   Find the reduced row echelon form.

   $$A=\left[\begin{array}{ccc}1& 2& 4\\ 3& 6& 12\end{array}\right]$$

   Answer: $\left[\begin{array}{ccc}1& 2& 4\\ 0& 0& 0\end{array}\right]$

   Step-by-step solution

   1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 2& 4\\ 0& 0& 0\end{array}\right]$$
   3. $$\text{Pivot columns: }\left(0,\right)$$
9. 9.

   Find the reduced row echelon form.

   $$A=\left[\begin{array}{ccc}2& -1& 1\\ 1& 2& 3\end{array}\right]$$

   Answer: $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\end{array}\right]$

   Step-by-step solution

   1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\end{array}\right]$$
   3. $$\text{Pivot columns: }\left(0,1\right)$$
10. 10.

    Find the reduced row echelon form.

    $$A=\left[\begin{array}{ccc}3& 6& 3\\ 1& 2& 1\end{array}\right]$$

    Answer: $\left[\begin{array}{ccc}1& 2& 1\\ 0& 0& 0\end{array}\right]$

    Step-by-step solution

    1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 2& 1\\ 0& 0& 0\end{array}\right]$$
    3. $$\text{Pivot columns: }\left(0,\right)$$
11. 11.

    Find the reduced row echelon form.

    $$A=\left[\begin{array}{ccc}1& 4& -2\\ 2& 1& 5\end{array}\right]$$

    Answer: $\left[\begin{array}{ccc}1& 0& \frac{22}{7}\\ 0& 1& -\frac{9}{7}\end{array}\right]$

    Step-by-step solution

    1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& \frac{22}{7}\\ 0& 1& -\frac{9}{7}\end{array}\right]$$
    3. $$\text{Pivot columns: }\left(0,1\right)$$
12. 12.

    Find the reduced row echelon form.

    $$A=\left[\begin{array}{ccc}1& 1& 3\\ 2& 3& 8\\ 1& 2& 5\end{array}\right]$$

    Answer: $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$

    Step-by-step solution

    1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$$
    3. $$\text{Pivot columns: }\left(0,1\right)$$
13. 13.

    Find the reduced row echelon form.

    $$A=\left[\begin{array}{ccc}2& 0& 4\\ 1& 1& 3\end{array}\right]$$

    Answer: $\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 1\end{array}\right]$

    Step-by-step solution

    1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 1\end{array}\right]$$
    3. $$\text{Pivot columns: }\left(0,1\right)$$
14. 14.

    Find the reduced row echelon form.

    $$A=\left[\begin{array}{ccc}1& -2& 1\\ 3& 1& 2\end{array}\right]$$

    Answer: $\left[\begin{array}{ccc}1& 0& \frac{5}{7}\\ 0& 1& -\frac{1}{7}\end{array}\right]$

    Step-by-step solution

    1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& \frac{5}{7}\\ 0& 1& -\frac{1}{7}\end{array}\right]$$
    3. $$\text{Pivot columns: }\left(0,1\right)$$
15. 15.

    Find the reduced row echelon form.

    $$A=\left[\begin{array}{ccc}4& 8& 4\\ 1& 2& 1\end{array}\right]$$

    Answer: $\left[\begin{array}{ccc}1& 2& 1\\ 0& 0& 0\end{array}\right]$

    Step-by-step solution

    1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 2& 1\\ 0& 0& 0\end{array}\right]$$
    3. $$\text{Pivot columns: }\left(0,\right)$$
16. 16.

    Find the reduced row echelon form.

    $$A=\left[\begin{array}{ccc}1& 0& 5\\ 0& 1& 2\\ 2& 1& 12\end{array}\right]$$

    Answer: $\left[\begin{array}{ccc}1& 0& 5\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$

    Step-by-step solution

    1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 5\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$$
    3. $$\text{Pivot columns: }\left(0,1\right)$$
17. 17.

    Find the reduced row echelon form.

    $$A=\left[\begin{array}{ccc}2& 3& 1\\ 4& 6& 2\end{array}\right]$$

    Answer: $\left[\begin{array}{ccc}1& \frac{3}{2}& \frac{1}{2}\\ 0& 0& 0\end{array}\right]$

    Step-by-step solution

    1. $$Useelementaryrowoperationstocleareachpivotcolumn.$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& \frac{3}{2}& \frac{1}{2}\\ 0& 0& 0\end{array}\right]$$
    3. $$\text{Pivot columns: }\left(0,\right)$$

More from Linear Algebra

• Vector Operations
• Dot Products
• Matrix Products
• Linear Systems
• Determinants
• Matrix Inverses
• Linear Independence
• Rank & Basis
• Orthogonal Projections
• Linear Transformations
• Eigenvalues