Math & Trigonometry
3x3 Matrix Determinant and Cramer's Rule Linear System Solver (MDETERM & MINVERSE)
Calculate exact 2x2 and 3x3 matrix determinants, test for singular zero-determinant systems, and solve simultaneous linear equations A*x = b via cofactor expansion.
Cofactor determinant expansion and Cramer's Rule
For a 3x3 coefficient matrix A in A2:C4, the determinant det(A) expands along row 2 as A2*(B3*C4 - C3*B4) - B2*(A3*C4 - C3*A4) + C2*(A3*B4 - B3*A4). Whenever det(A) = 0, the rows or columns are linearly dependent and the matrix inverse MINVERSE does not exist.
When det(A) is non-zero, Cramer's Rule solves A*x = b for each unknown x, y, z by replacing the corresponding column of A with the right-hand vector D2:D4 and dividing that modified determinant by det(A).
| Equation Row | x Coeff (A) | y Coeff (B) | z Coeff (C) | RHS Constant (D) |
|---|---|---|---|---|
| Eq 1 | 2 | 1 | -1 | 8 |
| Eq 2 | -3 | -1 | 2 | -11 |
| Eq 3 | -2 | 1 | 2 | -3 |
| Solution | x = 2 | y = 3 | z = -1 | det(A) = -1 |
3x3 Matrix Determinant Formula (MDETERM Equivalent)
Evaluates the exact cofactor expansion of A2:C4.
=A2 * (B3 * C4 - C3 * B4) - B2 * (A3 * C4 - C3 * A4) + C2 * (A3 * B4 - B3 * A4)Cramer's Rule Unknown Solver With Singularity Guard
Solves for x (or y, z) by dividing the RHS-substituted determinant by det(A), returning "Singular Matrix" when det(A) = 0.
=LET(det_a, A2 * (B3 * C4 - C3 * B4) - B2 * (A3 * C4 - C3 * A4) + C2 * (A3 * B4 - B3 * A4), det_x, D2 * (B3 * C4 - C3 * B4) - B2 * (D3 * C4 - C3 * D4) + C2 * (D3 * B4 - B3 * D4), IF(det_a = 0, "Singular Matrix", det_x / det_a))Try the example
Use the original values or change them to test your understanding.