Master Determinants: Cofactor Expansion, Triangular Matrices, and Cramer's Rule

1. Second‑order Determinant

For a 2×2 matrix \(\begin{pmatrix}a & b\\ c & d\end{pmatrix}\), the determinant is calculated as ad − bc .

2. Third‑order Determinant

For a 3×3 matrix, the determinant can be obtained by expanding along any row or column using the cofactor method.

3. Cofactor Expansion (Cofactor Matrix)

The cofactor of an element \(a_{ij}\) is \((-1)^{i+j}\) times the minor determinant obtained by deleting the i‑th row and j‑th column.

3.1 Example 1

Given a specific 3×3 matrix, the determinant is expressed as the sum of each element of the first row multiplied by its corresponding cofactor.

3.2 Example 2

A second example shows the expansion when a column contains two zeros, simplifying the calculation.

3.3 Cofactor Matrix

The cofactor matrix (also called the adjugate matrix) consists of all cofactors of the original matrix, arranged in the same positions.

4. Triangular Matrix

If a matrix is triangular (upper or lower), repeatedly applying cofactor expansion reduces the determinant to the product of the diagonal elements.

4.1 Example 3

Illustrations of an upper‑triangular and a lower‑triangular matrix confirm the rule.

5. Properties of Determinants

If a row (or column) of a square matrix is multiplied by a scalar \(k\), the determinant is multiplied by \(k\).

Swapping two rows (or columns) changes the sign of the determinant.

Adding a multiple of one row to another row leaves the determinant unchanged.

The determinant of a product of two square matrices equals the product of their determinants.

A square matrix is invertible if and only if its determinant is non‑zero.

For a scalar \(k\) and a square matrix \(A\), \(\det(kA)=k^{n}\det(A)\) where \(n\) is the order of the matrix.

If \(A\) is invertible, \(\det(A^{-1})=1/\det(A)\).

6. Cramer's Rule

For a linear system \(A\mathbf{x}=\mathbf{b}\) with a nonsingular square matrix \(A\), the solution for each variable \(x_i\) is given by \(x_i=\frac{\det(A_i)}{\det(A)}\), where \(A_i\) is obtained by replacing the i‑th column of \(A\) with the vector \(\mathbf{b}\).

7. Inverse Matrix via Adjugate

An alternative way to compute \(A^{-1}\) is \(A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)\), where \(\operatorname{adj}(A)\) is the transpose of the cofactor matrix.

8. Summary

The article covered determinant calculation using cofactor expansion, properties of determinants, special cases for triangular matrices, and applications such as Cramer's rule and the adjugate method for matrix inversion.