Solving Systems of Linear Equations with Matrix Form

Core Definition

Any system of simultaneous linear equations can be represented compactly as a single matrix-vector equation:

A\vec{x} = \vec{b}

where:

$A$ is the coefficient matrix containing the scalar multipliers of each variable.
$\vec{x}$ is the column vector of unknown variables.
$\vec{b}$ is the column vector of constant values.

When matrix $A$ is invertible (square with non-zero determinant), the exact solution is given by:

\vec{x} = A^{-1}\vec{b}

Concept Explanation

Translating Linear Equations into Matrix Form

Consider a two-variable system of linear equations:

\begin{cases} a_{11}t + a_{12}s = b_1 \\ a_{21}t + a_{22}s = b_2 \end{cases}

This translates directly into matrix multiplication:

\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} t \\ s \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}

Solving Efficiently with Matrix Calculators

In modern engineering practice, computing solutions to linear systems by hand via substitution or elimination can be slow and prone to arithmetic sign slips.

By expressing the system in matrix form $A\vec{x} = \vec{b}$ , engineers solve for the variable vector $\vec{x}$ by multiplying both sides by the matrix inverse:

\vec{x} = A^{-1}\vec{b}

Using modern digital tools like the Desmos Matrix Calculator or scientific graphing calculators:

Define Matrix $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$
Define Column Matrix $B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$
Command the calculator to evaluate: $A^{-1} B$ The output matrix immediately returns the exact scalar values for the unknowns $\begin{bmatrix} t \\ s \end{bmatrix}$ .

Overdetermined Systems in 3D Geometry ( $3$ Equations, $2$ Unknowns)

When analyzing 3D lines in space (e.g. testing if two trajectories intersect in Relationships Between Lines in Space), we encounter systems with 3 coordinate equations ( $x, y, z$ ) and 2 parameter unknowns ( $t, s$ ):

\begin{cases} x_1(t) = x_2(s) \implies c_1 t + d_1 s = k_1 \\ y_1(t) = y_2(s) \implies c_2 t + d_2 s = k_2 \\ z_1(t) = z_2(s) \implies c_3 t + d_3 s = k_3 \end{cases}

The 2-Step Matrix Verification Workflow:

Solve the $2 \times 2$ Subsystem: Select any two equations (e.g., equations 1 and 2), set up $A = \begin{bmatrix} c_1 & d_1 \\ c_2 & d_2 \end{bmatrix}$ and $B = \begin{bmatrix} k_1 \\ k_2 \end{bmatrix}$ , and compute $\begin{bmatrix} t \\ s \end{bmatrix} = A^{-1} B$ using a matrix calculator.
Verify in the 3rd Equation (Consistency Check): Substitute the computed values of $t$ and $s$ into the remaining 3rd equation ( $z$ -component):
- Consistent ( $\text{Left Side} = \text{Right Side}$ ): The lines share a common point and intersect.
- Inconsistent ( $\text{Left Side} \neq \text{Right Side}$ ): The system has no solution; the non-parallel lines are skew.

Engineering Applications

3D Collision & Ray Tracing: Determining whether line-of-sight vectors intersect geometric mesh boundaries in computer graphics.
Structural Truss Statics: Solving simultaneous node equilibrium equations ( $\sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ ) for member tension and compression forces.
Electrical Circuit Analysis: Solving Kirchhoff's Current Law (KCL) matrix systems across interconnected circuit meshes.

Example Problems

Example 1: Solving a $2 \times 2$ Parameter System via Matrix Form

Solve the linear parameter system:

\begin{cases} 5t + 6s = -1 \\ -6t + 12s = 1 \end{cases}

Solution

Write in Matrix Form $A\vec{x} = \vec{b}$ : $\begin{bmatrix} 5 & 6 \\ -6 & 12 \end{bmatrix} \begin{bmatrix} t \\ s \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \end{bmatrix}$
Formulate the Matrix Solution: $\begin{bmatrix} t \\ s \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ -6 & 12 \end{bmatrix}^{-1} \begin{bmatrix} -1 \\ 1 \end{bmatrix}$
Evaluate via Matrix Calculator: Entering matrix $A$ and column vector $B$ into a matrix calculator yields: $\begin{bmatrix} t \\ s \end{bmatrix} = \begin{bmatrix} -\frac{3}{16} \\ -\frac{1}{96} \end{bmatrix}$ Therefore, $t = -\frac{3}{16}$ and $s = -\frac{1}{96}$ .

Connections & References

Parent Concepts: Vectors in 2D and 3D | Equations of Lines in 3D
Related Notes: Relationships Between Lines in Space | Dot and Cross Products
Course Sessions: Calc-3 Session 04