Number of Solutions from a Matrix
This page follows the JoeCMath video The Matrix Gives Away the Answer. The video shows how the final row-reduced augmented matrix tells you whether a system has one solution, infinitely many solutions, or no solution.
Main Idea
After Gaussian elimination or Gauss-Jordan elimination, the final augmented matrix usually gives away the type of answer.
The video sorts the final matrix into three outcomes:
- one solution,
- infinitely many solutions,
- no solution.
The key is to read the final row and the pivot columns. A pivot for every variable points toward one solution. A row of all zeroes on the coefficient side can create a free variable. A row that says $0$ equals a nonzero number means no solution.
Watch this section: overview of the three outcomes at 0:08.
One Solution
The unique-solution pattern in the video is a row-reduced matrix shaped like
\[\left[ \begin{array}{ccc|c} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \end{array} \right].\]Each variable column has a pivot. That means each variable is locked to one value:
\[\begin{aligned} x&=a, \\ y&=b, \\ z&=c. \end{aligned}\]So the solution is the single ordered triple
\[(x,y,z)=(a,b,c).\]Watch this section: unique solution example at 1:37.
Reading a Gauss-Jordan Result
When the matrix is in reduced row echelon form, the answer can be read directly. The left side looks like an identity matrix, and the right side gives the values of the variables.
For example,
\[\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 5 \end{array} \right]\]means
\[\begin{aligned} x&=2, \\ y&=-1, \\ z&=5. \end{aligned}\]That is why the video treats a fully reduced Gauss-Jordan result as the cleanest final form for reading one solution.
Watch this section: finding the unique solution from a Gauss-Jordan result at 3:30.
Reading a Gaussian Result
A Gaussian elimination result may stop in row echelon form instead of reduced row echelon form. In that case, the matrix may not give every variable directly from left to right.
The idea is still the same: start with the bottom pivot equation, solve that variable first, then substitute upward. The final matrix may have a triangular shape like
\[\left[ \begin{array}{ccc|c} 1 & * & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \end{array} \right].\]In that form, the last row solves the last variable first. Then the row above it can use that value, and the top row is handled last. This is the back-substitution part of reading a Gaussian elimination result.
Watch this section: finding the unique solution from a Gaussian result at 3:52.
Check the Solution
The video also emphasizes that the values should satisfy the original system. After you read a solution from the final matrix, plug the values back into the original equations.
If the matrix leads to
\[(x,y,z)=(a,b,c),\]then each original equation should become a true statement after substituting $x=a$, $y=b$, and $z=c$.
This check is especially useful when the row-reduction work has several arithmetic steps. The final matrix tells the story, but substitution confirms that the story is consistent.
Watch this section: verifying the solution in the original system at 4:59.
Infinitely Many Solutions
The infinite-solution pattern in the video has at least one free variable and no contradiction row. A typical reduced matrix can look like
\[\left[ \begin{array}{ccc|c} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 0 & 0 \end{array} \right].\]The last row says
\[0=0,\]which is true but does not solve for another variable. If there are three variables and only two pivot variables, then one variable is free.
Watch this section: infinite solutions example at 5:36.
Free Variables
A free variable is a variable without a pivot. The video uses this idea to describe infinitely many solutions as a set of solutions rather than one ordered triple.
For example, if $z$ is free, you can write
\[z=t.\]Then the pivot variables can be written in terms of $t$. A solution set might have the shape
\[(x,y,z)=(a,b,t),\]Different choices of $t$ give different ordered triples, so there are infinitely many solutions.
Watch this section: free variable discussion at 6:33.
No Solution
The no-solution pattern is the contradiction row. In the video thumbnail, the final row has zeroes on the coefficient side but a nonzero number on the constants side:
\[\left[ \begin{array}{ccc|c} 1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 0 & 5 \end{array} \right].\]The last row means
\[0x+0y+0z=5.\]That simplifies to
\[0=5,\]which is impossible. Since one row of the system is impossible, the whole system has no solution.
Watch this section: no solution example at 9:28.
Timestamp Guide
Related Linear Algebra Work
This video fits naturally after the Augmented Matrices companion, Row Echelon Form companion, Reduced Row Echelon Form companion, and Gauss-Jordan Elimination companion.
For more row-reduction lessons, use the JoeCMath Gauss-Jordan Elimination playlist.