Cramer's Rule Formula

Linear Algebra (Matrices) · Axiom Academy

Each unknown is a ratio of two determinants — and the only thing that changes between them is one swapped column. 1. Each Unknown Is a Ratio of Determinants Take a square system — the same number of equations as unknowns. If the coefficient determinant is not zero , Cramer's Rule hands you each unknown directly, with no row reduction: one determinant over another — that shared denominator is 2. Building A_i : Replace Column i With Where does the numerator come from? Start with the coefficient matrix A and the constants . To solve for the i -th unknown, slide in and let it replace column i of A — that new matrix is A_i . Watch sweep across columns 1 , 2 , 3 in turn, each time producing one unknown as a determinant ratio: For x_1 : replace column 1 with to get A_1 , then . For x_2 : replace column 2 with to get A_2 , then . For x_i in general: the denominator is fixed; you re-aim at column i and recompute only the numerator. Every unknown is divided by , so the rule simply cannot run when that denominator is zero — you would be dividing by 0 . Geometrically, means the columns of A collapse onto the same line: they stop spanning the plane, and the system loses its unique solution. Watch the denominator shrink toward zero and the ratio blow up: Each ratio is a finite number — exactly one solution, and Cramer's Rule delivers it. The columns collapse onto one line.

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