Chapter 3

3.1 Introduction to Determinants

For n >= 2, the determinant of an n x n matrix A = [a_ij] is the sum of n terms of the form +-a_1j det A_1j, with plus and minus signs alternating, where the entries a₁₁, a₁₂, ..., a_1n are from the first rows of A. In symbols, det A = a₁₁ det A₁₁ - a₁₂ det A₁₂ + ... + (-1)⁽¹⁺ⁿ⁾ a_1n det A_1n = sum from j = 1 to n of (-1)^(1+j) a_1j det A_1j.

Given A = [a_ij], the (i, j) - cofactor of A is the number C_ij given by C_ij = (-1)^(i+j) det A_ij. The determinant of an n x n matrix A can be computed by a cofactor expansion across any row or down any column. The expansion across the ith row is det A = a_i1C_i1 + a_i2C_i2 + ... + a_inC_in. The expansion across the jth column is det A = a_1jC_1j + a_2jC_2j + ... + a_njC_nj.

If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A.

3.2 Properties of Determinants

Row Operations
Let A be a square matrix.
a. If a multiple of one row of A is added to another row to produce a matrix B, then det B = det A.
b. If two rows of A are interchanged to produce B, then det B = -det A.
c. If one row of A is multiplied by k to produce B, then det B = k x det A.

A square matrix A is invertible if and only if det A is not equal to 0.

If A is an n x n matrix, then det A^T = det A.

If A and B are n x n matrices, then det(AB) = det(A) x det(B).