Training Linear Algebra Systems and Row Reduction
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Systems and Row Reduction

24 min Linear Algebra

Systems and Row Reduction

Systems of linear equations can be organized into an augmented matrix and solved with row operations.

Elementary Row Operations
  1. Swap two rows.
  2. Multiply a row by a nonzero scalar.
  3. Add a multiple of one row to another.
Example 1

Solve $x+y=5$ and $x-y=1$.

Augmented matrix: $$\left[\begin{array}{cc|c}1&1&5\1&-1&1\end{array}\right].$$ Subtract row 2 from row 1 or solve directly to get $(x,y)=(3,2)$.

Reduced Row-Echelon Form

A matrix is in RREF when each leading entry is 1, leading 1s move right as you go down, and each leading 1 is the only nonzero entry in its column.

Example 2

Interpret $$\left[\begin{array}{cc|c}1&0&4\0&1&-2\end{array}\right].$$

$x=4$, $y=-2$.

Special Cases

A row like $[0\;0\;|\;5]$ means no solution. A row of all zeros can signal infinitely many solutions.

Practice Problems

1. Write the augmented matrix for $2x+y=7$, $x-y=1$.
2. Solve $x+y=6$, $x-y=2$.
3. What does a contradictory row look like?
4. What does a zero row suggest?
5. Interpret $$\left[\begin{array}{cc|c}1&0&-1\0&1&3\end{array}\right].$$
Show Answer Key

1. $$\left[\begin{array}{cc|c}2&1&7\1&-1&1\end{array}\right]$$

2. $(4,2)$

3. $[0\;0\;|\;c]$ with $c \ne 0$

4. Dependent equations or a free variable

5. $(x,y)=(-1,3)$

Matrices and Matrix Operations Determinants and Inverses