n×n determinants: cofactors & Cramer's rule

Japanese school year: University year 1

What you learn

Learn the properties of general n-th order determinants, calculation via cofactor expansion, and Cramer's rule. These tools are widely applied to solve linear systems with many unknowns and to formulate eigenvalue problems. Prior proficiency in evaluating 2x2 determinants and performing basic matrix operations is required.

Go to practice

Key points

This formula (Laplace expansion) calculates a large determinant by breaking it down into smaller determinants $M_{ij}$ (minors). Choose row $i$ , multiply each entry $a_{ij}$ by its alternating sign and corresponding minor $M_{ij}$ , then sum them up.

\det A = \sum_{j=1}^n (-1)^{i+j} a_{ij} M_{ij}

Cramer's rule solves for the unknown $x_i$ in a linear system directly using determinants. You divide by $\det A$ , where matrix $A_i$ is formed by replacing column $i$ of matrix $A$ with the constants from the right-hand side.

x_i = \frac{\det A_i}{\det A} \quad (\det A \neq 0)

This property shows that transposing matrix $A^T$ (swapping rows and columns) preserves the determinant of matrix $A$ . It means that any calculation rule that applies to rows also works for columns.

\det(A^T) = \det A

Choose a set to practice.

Practice 1Coming soon