Matrices & linear systems

augmented matrix

An augmented matrix is a bookkeeping trick for solving a linear system A*x = b. You glue the coefficient matrix A and the right-hand-side vector b together into one wider grid, often written [A | b] with a bar marking where the equals signs were.

The point is convenience. When you run elimination, every row operation has to be applied to both sides of each equation. Packing them into one grid means a single sweep of row operations handles A and b together, with no chance of forgetting the right-hand side.

For x + y = 5 and x - y = 1, the augmented matrix is [[1,1,5],[1,-1,1]]. Reduce it to echelon form and the solution can be read straight off the final grid.

x + y = 5, x - y = 1 -> [A | b] = [[1, 1 | 5], [1, -1 | 1]]

Coefficients and answers in one grid for row reduction.

The bar is only a visual reminder; the row operations treat the whole grid as one.

Also called

[A | b]augmented coefficient matrix增广矩阵增廣矩陣增广系数矩阵