Cramers Rule for Multivariable Systems

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What is Cramer's Rule?

An explicit algebraic formula for solving a system of linear equations with as many equations as unknowns, where each variable is expressed as the ratio of two determinants: $x_i = \frac{D_i}{D}$.

In Cramer's Rule for a $2 \times 2$ system, how is the numerator determinant $D_x$ formed?

Take the coefficient matrix determinant $D$ and replace the $x$-coefficients column with the column of constants from the right-hand side of the equations.

State the explicit formulas for $x$ and $y$ using Cramer's Rule for the system: $$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$$

$$ x = \frac{D_x}{D} = \frac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}, \quad y = \frac{D_y}{D} = \frac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}} $$ (provided $D \neq 0$).

Solve for $y$ using Cramer's Rule: $$\begin{cases} 2x + 3y = 8 \\ 4x - y = 2 \end{cases}$$

1. $D = \begin{vmatrix} 2 & 3 \\ 4 & -1 \end{vmatrix} = (2)(-1) - (3)(4) = -2 - 12 = -14$. 2. Replace $y$-column with constants: $$D_y = \begin{vmatrix} 2 & 8 \\ 4 & 2 \end{vmatrix} = (2)(2) - (8)(4) = 4 - 32 = -28$$ 3. $y = \frac{D_y}{D} = \frac{-28}{-14} = 2$.

When does Cramer's Rule fail to produce a unique solution, and how are the outcomes distinguished?

Cramer's Rule fails when the main determinant $D = 0$: - If $D = 0$ and at least one numerator determinant $D_i \neq 0$, the system is inconsistent (no solution). - If $D = 0$ and all numerator determinants $D_i = 0$, the system is typically dependent (infinitely many solutions).

How is Cramer's Rule applied to find variable $z$ in a $3 \times 3$ system?

$$ z = \frac{D_z}{D} $$ where $D$ is the $3 \times 3$ determinant of coefficients and $D_z$ is formed by replacing the third column (the $z$-coefficients) with the constant terms column.

Why is Gaussian elimination generally preferred over Cramer's Rule for larger systems (e.g., $4 \times 4$ or greater)?

Calculating determinants of large matrices is computationally expensive (factorial growth via cofactor expansion), whereas Gaussian elimination scales polynomially ($O(n^3)$), making it far faster and less prone to arithmetic errors.

Cramers Rule for Multivariable Systems

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