What is an Augmented Matrix?
An augmented matrix is a convenient way to represent a system of linear equations in a single matrix structure. Instead of writing out variables like x, y, and z repeatedly, we write only the coefficients in a grid. We then append (augment) a final column containing the constants from the right-hand side of the equal signs. A vertical bar is typically drawn in the matrix to separate the coefficients from the constants:
| a11 | a12 | a13 | b1 |
| a21 | a22 | a23 | b2 |
| a31 | a32 | a33 | b3 |
Solving Linear Systems Using Augmented Matrices
To solve the system of equations represented by the augmented matrix, we perform Gauss-Jordan elimination on the entire matrix. Any row operation performed on the coefficient side must also be performed on the constant side. This keeps the equations balanced.
Once the augmented matrix is reduced to Reduced Row Echelon Form (RREF), the solutions can be read directly from the final column:
- If the coefficient side reduces to the Identity matrix, we have a unique solution, and the values are listed in the final column.
- If we get a row of all zeros on the left, but a non-zero value on the right (e.g., [0 0 0 | 5]), the system is inconsistent and has no solution.
- If we get a row of all zeros on both sides (e.g., [0 0 0 | 0]) and there are columns without pivots, the system is consistent and has infinitely many solutions.
Augmenting is also used to compute matrix inverses by creating the matrix [A | I], where A is reduced to the identity and I becomes the inverse.