Business Mathematics and Statistics · Ch 1 — Applications of Matrices and Determinants (Rank, Cramer's Rule, Transition Probability Matrices)

Cramer's Rule for Three Variables

The formulas

For the system

a_1x+b_1y+c_1z=d_1,\quad a_2x+b_2y+c_2z=d_2,\quad a_3x+b_3y+c_3z=d_3

form the coefficient determinant $\Delta$ and three more determinants $\Delta_x,\Delta_y,\Delta_z$ , each obtained from $\Delta$ by replacing ONE column with the constants column:

\Delta=\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}\qquad \Delta_x=\begin{vmatrix}d_1&b_1&c_1\\d_2&b_2&c_2\\d_3&b_3&c_3\end{vmatrix}

\Delta_y=\begin{vmatrix}a_1&d_1&c_1\\a_2&d_2&c_2\\a_3&d_3&c_3\end{vmatrix}\qquad \Delta_z=\begin{vmatrix}a_1&b_1&d_1\\a_2&b_2&d_2\\a_3&b_3&d_3\end{vmatrix}

Provided $\Delta\neq0$ (the equations are independent — exactly $\rho(A)=3$ from the previous section), the unique solution is

x=\frac{\Delta_x}{\Delta}\qquad y=\frac{\Delta_y}{\Delta}\qquad z=\frac{\Delta_z}{\Delta}

Note

Only one column changes at a time

$\Delta_x$ replaces only the FIRST column ( $x$ -coefficients), $\Delta_y$ only the SECOND, $\Delta_z$ only the THIRD. Replacing the wrong column, or more than one at once, is the single most common Cramer's Rule setup mistake. …

Definition 1Cramer's Rule (Three Variables)

For $a_1x+b_1y+c_1z=d_1$ etc. with $\Delta\neq0$ : $x=\Delta_x/\Delta$ , $y=\Delta_y/\Delta$ , $z=\Delta_z/\Delta$ , where each subscripted determinant replaces the matching column of $\ …

Start your 14-day free trial to unlock the full solution →

← Previous · 2Consistency of a System of Linear Equations — the Rank Method Next · 4 →Transition Probability Matrices (Markov Chains)