Lesson 29 — 2x2 and 3x3 Linear Systems

Methods and theory

Solution methods

1. Substitution: isolate one variable and substitute into the other.
2. Addition (elimination): combine equations to eliminate a variable.
3. Cramer: ratio of determinants.
4. Row reduction (Gauss): triangularize the matrix.
5. Inverse matrix: $\mathbf x = A^{-1}\mathbf b$.

Cramer 2x2

For $\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$ with $D = a_1 b_2 - a_2 b_1 \neq 0$:

$x = \frac{c_1 b_2 - c_2 b_1}{D}, \quad y = \frac{a_1 c_2 - a_2 c_1}{D}$

Cramer 3x3

3x3 determinant (Sarrus): $\det A = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$

$x_i = \frac{\det A_i}{\det A}$

where $A_i$ has the $i$-th column replaced by $\vec b$.

Classification by determinant

Row reduction (Gauss)

Elementary operations:

1. Swap two equations.
2. Multiply an equation by a non-zero scalar.
3. Add a multiple of one equation to another.

Goal: upper triangular. Then back-substitution.