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Algebra2D

Matrices and Determinants - Theory & Concepts - Cramers Rule

Comprehensive guide to matrix operations, calculating determinants, finding inverses, and using Cramer's Rule.

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Cramer's Rule 2x2 Matrix Solver

Equation 1

2x+(1)y=42x + (1)y = 4
a₁ = 2
b₁ = 1
c₁ = 4

Equation 2

1x+(−1)y=11x + (-1)y = 1
a₂ = 1
b₂ = -1
c₂ = 1

Cramer's Step-by-Step Determinants

Main D

D=∣211−1∣=−3D = \begin{vmatrix} 2 & 1 \\ 1 & -1 \end{vmatrix} = -3

Dx (Replace x)

Dx=∣411−1∣=−5D_x = \begin{vmatrix} 4 & 1 \\ 1 & -1 \end{vmatrix} = -5

Dy (Replace y)

Dy=∣2411∣=−2D_y = \begin{vmatrix} 2 & 4 \\ 1 & 1 \end{vmatrix} = -2
6-66-6
Eq 1 (Cyan)Eq 2 (Purple)

System Solution

Unique Solution:

x=DxD=−5−3=1.667x = \frac{D_x}{D} = \frac{-5}{-3} = 1.667y=DyD=−2−3=0.667y = \frac{D_y}{D} = \frac{-2}{-3} = 0.667

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