Business Mathematics and Basic Statistics · Ch 11 — Determinants and Cramer's Rule

Setting Up Cramer's Rule — the Determinants D, Dₓ and D_y

The system this chapter solves

This chapter uses determinants to solve a system of two linear equations in two unknowns, $x$ and $y$ :

a_1x+b_1y=c_1

a_2x+b_2y=c_2

with exactly two independent equations for the two unknowns — the scope of this syllabus does not extend to a system with more or fewer independent equations than unknowns.

The three determinants

Cramer's Rule is built from THREE determinants, each formed from the system's own coefficients:

D = \begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix} \qquad D_x = \begin{vmatrix}c_1&b_1\\c_2&b_2\end{vmatrix} \qquad D_y = \begin{vmatrix}a_1&c_1\\a_2&c_2\end{vmatrix}

$D$ is the determinant of the coefficients of $x$ and $y$ , taken exactly as they appear in the two equations.
$D_x$ is $D$ with its FIRST column (the $x$ -coefficients) replaced by the constants $c_1,c_2$ .
$D_y$ is $D$ with its SECOND column (the $y$ -coefficients) replaced by the constants $c_1,c_2$ .

Note

Only one column changes at a time

$D_x$ replaces ONLY the $x$ -coefficient column, leaving the $y$ -coefficient column untouched; $D_y$ replaces ONLY the $y$ -coefficient column, leaving the $x$ -coefficient column untouched. Replacing the wrong column, or both columns at once, is the most common setup mistake with Cramer's Rule.

When does a unique solution exist? …

Definition 1D (Coefficient Determinant)

The determinant formed from the x- and y-coefficients of the system, exactly as they appear in t …

Definition 2D_x

D with its first (x-coefficient) column replaced by the constants on the right-hand side of t …

Definition 3D_y

D with its second (y-coefficient) column replaced by the constants on the right-hand side of t …

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