Lesson 35 — Solving Systems via Matrices

Solution methods

Matrix form

$\begin{cases} a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3 \end{cases}$ ⟺ $A\mathbf{x} = \mathbf{b}$ with $A_{3 \times 3}$, $\mathbf{x}, \mathbf{b} \in \mathbb{R}^3$.

Method 1 — Gaussian elimination

Elementary operations (do not change solution):

1. Swap two rows.
2. Multiply a row by a non-zero scalar.
3. Add a multiple of one row to another.

Goal: triangularize $[A | \mathbf{b}]$ to row-echelon form. Then back-substitute.

Method 2 — Cramer

For $A\mathbf{x} = \mathbf{b}$ with $\det A \neq 0$: $x_i = \frac{\det A_i}{\det A}$

where $A_i$ is $A$ with the $i$-th column replaced by $\mathbf{b}$.

Method 3 — Inverse

$\mathbf{x} = A^{-1} \mathbf{b}$. $A^{-1}$ can be computed via $[A | I] \to [I | A^{-1}]$ by row reduction.

When to use each

Classification of systems

• Determinate (unique): unique solution ($\det A \neq 0$, or $\text{rank}(A) = \text{rank}([A|\mathbf b]) = n$).
• Indeterminate: infinitely many solutions ($\text{rank}(A) = \text{rank}([A|\mathbf b]) < n$).
• Impossible: no solution ($\text{rank}(A) < \text{rank}([A|\mathbf b])$).

Rouché-Capelli theorem

$A\mathbf x = \mathbf b$ has a solution ⟺ $\text{rank}(A) = \text{rank}([A|\mathbf b])$. Solution is unique if both equal the number of unknowns.