No Solution or Infinitely Many Solutions in Matrices

When solving a system of equations using matrices, there can be a unique solution, infinitely many solutions, or no solution.
In row-reduced form, the matrix (if it has a unique solution) will consist of a diagonal of 1's surrounded entirely by zeros, and a column of constants to the far right; here is an example:

| 1  0  0 | 4 |
| 0  1  0 | 2 |
| 0  0  1 | 7 |

A matrix with infinitely many solutions has more variables than non-zero rows; here is an example:

| 1  0  3 | 4 |
| 0  1 -2 | 3 |
| 0  0  0 | 0 |

Finally, a matrix with no solution will have a row of zeros on the left side, while the constant on the right side is not 0. A matrix with no solution is called inconsistent; here is an example:

| 1  0  3 | 4 |
| 0  1 -2 | 3 |
| 0  0  0 | 2 |

When graphed, the three types of solutions will look similar to the following image, where consistent and independent means unique solution, consistent and dependent means infinitely many solutions, and inconsistent means no solution.