Determinants Demystified: Cofactor Expansion, Triangular Matrices & Cramer's Rule

1. Second‑Order Determinant

For a 2×2 matrix \(\begin{pmatrix}a & b\\ c & d\end{pmatrix}\), the determinant is computed 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 cofactors.

3. Cofactor Expansion

The cofactor of an element is the signed minor determinant obtained by deleting its row and column. The determinant equals the sum of elements of any row (or column) multiplied by their cofactors.

3.1 Example 1

Consider matrix A. Its cofactor expansion along the first row yields the determinant as the sum of each element times its corresponding cofactor.

3.2 Example 2

Apply cofactor expansion to a matrix that contains two zero entries in a column, simplifying the calculation.

3.3 Cofactor Matrix

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

4. Triangular Matrix

If a matrix is triangular, repeatedly applying cofactor expansion shows that its determinant equals the product of the diagonal elements.

4.1 Example 3

Upper‑triangular and lower‑triangular matrices illustrate this property.

5. Properties of Determinants

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

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

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

The determinant of a product equals the product of the determinants.

A square matrix is invertible iff its determinant is non‑zero.

Determinant of a transpose equals the determinant of the original matrix.

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

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

6. Cramer's Rule

For a linear system \(A\mathbf{x}=\mathbf{b}\) with invertible square matrix \(A\), replace the \(i\)‑th column of \(A\) with \(\mathbf{b}\) to form \(A_i\). Then \(x_i = \det(A_i)/\det(A)\).

7. Another Method for the Inverse Matrix

The inverse of \(A\) can be expressed as \(A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)\), where \(\operatorname{adj}(A)\) is the transpose of the cofactor matrix (the adjugate).

8. Summary

This article introduced determinant calculation via cofactor expansion, properties of determinants, simplifications for triangular matrices, Cramer's rule, and an adjugate‑based formula for matrix inversion.