Solving Systems of Equations with Matrices

Matrices provide a systematic and powerful method for solving systems of linear equations. This lesson covers expressing systems in matrix form, solving using the inverse matrix, and interpreting cases where the system has no solution or infinitely many solutions.

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Writing Systems in Matrix Form

A system of linear equations:

$$\begin{aligned} a_1 x + b_1 y + c_1 z &= d_1 \\ a_2 x + b_2 y + c_2 z &= d_2 \\ a_3 x + b_3 y + c_3 z &= d_3 \end{aligned}$$a1​x+b1​y+c1​za2​x+b2​y+c2​za3​x+b3​y+c3​z​=d1​=d2​=d3​​

can be written as $A\mathbf{x} = \mathbf{d}$Ax=d where:

$$A = \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{d} = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}$$A=​a1​a2​a3​​b1​b2​b3​​c1​c2​c3​​​,x=​xyz​​,d=​d1​d2​d3​​​

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Solving Using the Inverse

If $\det(A) \neq 0$det(A)=0, the system has a unique solution:

$\mathbf{x} = A^{-1}\mathbf{d}$x=A−1d

Worked Example 1: Solve:

$$\begin{aligned} 2x + y &= 5 \\ 3x + 2y &= 8 \end{aligned}$$2x+y3x+2y​=5=8​

Matrix form: $\begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 8 \end{pmatrix}$(23​12​)(xy​)=(58​)

$\det(A) = 4 - 3 = 1$det(A)=4−3=1. $A^{-1} = \begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix}$A−1=(2−3​−12​).

$$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix}\begin{pmatrix} 5 \\ 8 \end{pmatrix} = \begin{pmatrix} 10 - 8 \\ -15 + 16 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$$(xy​)=(2−3​−12​)(58​)=(10−8−15+16​)=(21​)

Solution: $x = 2, y = 1$x=2,y=1.

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The Three Cases

Determinant	Interpretation	Geometric meaning
$\det(A) \neq 0$det(A)=0	Unique solution	Lines/planes meet at one point
$\det(A) = 0$det(A)=0, consistent	Infinitely many solutions	Lines/planes coincide or intersect in a line
$\det(A) = 0$det(A)=0, inconsistent	No solution	Lines/planes are parallel

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Geometric Interpretation (3 Equations in 3 Unknowns)

Each equation represents a plane in 3D. The solution set is the intersection of three planes:

Configuration	Description	Solutions
Unique point	Three planes meet at a single point	One
Line	Three planes intersect along a line	Infinitely many (1 parameter)
Sheaf	Three planes meet along a common line	Infinitely many
Coincident planes	Two or more planes are identical	Infinitely many
Triangular prism	Planes form a prism (pairwise intersections are parallel lines)	None
Parallel planes	Two or more planes are parallel and distinct	None

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Worked Example 2: 3×3 System

Solve:

$$\begin{aligned} x + 2y + z &= 4 \\ 2x + y - z &= 3 \\ x + y + z &= 3 \end{aligned}$$x+2y+z2x+y−zx+y+z​=4=3=3​

$A = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 1 & -1 \\ 1 & 1 & 1 \end{pmatrix}$A=​121​211​1−11​​

$\det(A) = 1(1+1) - 2(2+1) + 1(2-1) = 2 - 6 + 1 = -3 \neq 0$det(A)=1(1+1)−2(2+1)+1(2−1)=2−6+1=−3=0

Since the determinant is non-zero, there is a unique solution. Finding $A^{-1}$A−1 and computing $A^{-1}\mathbf{d}$A−1d gives the solution.