47. Augmented Matrix

When solving the linear system $𝐴\overrightarrow{𝑥}=\overrightarrow{𝑏}$, the augmented matrix $\left[𝐴|\overrightarrow{𝑏}\right]$ stacks the coefficient matrix and the right-hand side into one $𝑚\times \left(𝑛+1\right)$ matrix:

The vertical bar separates the coefficients from the constants — purely visual; row operations treat it as one matrix.

47.1. Why use it

Row-reducing $\left[𝐴|\overrightarrow{𝑏}\right]$ to REF (via Gaussian elimination) or RREF (via Gauss–Jordan) reads off the solution directly — no need to substitute back manually if you go all the way to RREF.

47.2. Reading solutions

After row reduction:

• Unique solution: $\text{rank}\left(𝐴\right)=\text{rank}\left(\left[𝐴|\overrightarrow{𝑏}\right]\right)=𝑛$
• Infinite solutions: $\text{rank}\left(𝐴\right)=\text{rank}\left(\left[𝐴|\overrightarrow{𝑏}\right]\right)<𝑛$ (free variables)
• No solution: $\text{rank}\left(𝐴\right)<\text{rank}\left(\left[𝐴|\overrightarrow{𝑏}\right]\right)$ — a row $\left[0,0,\dots ,0|𝑐\right]$ with $𝑐\ne 0$ appears

This is the Rouché–Capelli theorem expressed in terms of augmented-matrix ranks.

47.3. Connections

• Gaussian Elimination — the algorithm that operates on $\left[𝐴|𝑏\right]$
• RREF — the canonical row-reduced form
• Homogeneous System — when $\overrightarrow{𝑏}=\mathrm{𝟎}$
• Linear System Solutions