Problem 58 - Algebra and Trigonometry

Question

When expanding a determinant by minors, when is it necessary to supply minus signs?

Step-by-Step Solution

Answer

When expanding a determinant by minors, a minus sign is necessary whenever the position indices (row number + column number) of the element under consideration add up to an odd number. This follows the rule \((-1)^{i+j}\) where 'i' is the row number and 'j' is the column number of the element.

Step 1: Understand determinant properties

Determinants are mathematical objects derived from a square matrix. Properties of determinants imply certain rules, including the rule of sign which determines the sign of each element of the matrix in a determinant's expansion. The sign of an element depends on the position of the element within the matrix.

Step 2: Reveal the rule of signs

The rule of signs is actually based on the positions or indices of the elements of a matrix. The element in the i-th row and j-th column of the matrix follows the rule of \((-1)^{i+j}\). When \(i+j\) is even, the sign will be positive; when \(i+j\) is odd, the sign will be negative.

Step 3: Connect to cofactor expansion

Therefore, while expanding a determinant, a minus sign is needed whenever the element's position indices add up to an odd number. This is crucial in the calculation of the cofactor, which is used in the process of expansion by minors.