all al2 al3 a2l a22 a23 a3l a32 a33


| A |= ana22<ï33 — ana23<ï32 + «12^23^31 — «12^21^33 + aoa2ia32 — aoa22a3i A careful examination of the evaluation of a 3 x 3 determinant shows:

1. Each term in the expansion of the determinant contains one and only one element from each row and each column.

2. The number of elements in each term is the same as the number of rows (or columns) in the matrix. Thus, a 2 x 2 determinant has two elements


in each term of its expansion, a 3 x 3 determinant has three elements in each term of its expansion, and so on.

3. The terms in the expansion alternate in sign from + to —.

4. A 2 x 2 determinant has two terms in its expansion, and a 3 x 3 determinant has six terms in its expansion. The general rule is: The determinant of order N x N has N! = N(N — 1)(N — 2) ••• 3 ■ 2 ■ 1 terms in its expansion, where N! is read "N factorial." Following this rule, a determinant of order 5 x 5 will have 5 ■ 4 ■ 3 ■ 2 ■ 1 = 120 terms in its expansion.1

Properties of Determinants

1. A matrix whose determinantal value is zero is called a singular matrix, whereas a matrix with a nonzero determinant is called a nonsingular matrix. The inverse of a matrix as defined before does not exist for a singular matrix.

2. If all the elements of any row of A are zero, its determinant is zero. Thus,

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