Problem 58 - College Algebra

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, it is necessary to supply minus signs for the elements on the odd positions (1st, 3rd, etc.) of the matrix while calculating the cofactors. The resulting 'chessboard' of signs begins positive from the top-left, and alternates between positive and negative both row-wise and column-wise. This is because in matrix, sign is determined by the sum of row and column coordinates (both starting from 1).

Step 1: Understanding Minors and Cofactors

In determinant expansion, a minor of an element in a determinant is created by removing the row and column containing that element. The cofactor is then calculated as the product of the minor and (-1) raised to the sum of the row number and column number (both starting from 1) of the element.

Step 2: Applying Sign Rule

When expanding a determinant, the sign associated with each term is determined by its position. In a standard square matrix, the top left element is given a positive sign. From there, sign alternates row-wise and column-wise; positive sign for elements on even places (2nd, 4th,...) and negative sign for elements on odd places (1st, 3rd,...).

Step 3: Understanding the Reason Behind Sign Rule

The reason behind the alternating signs comes from properties of vector spaces and linear independence. This rule ensures that redundant information (linearly dependent rows or columns) will cancel out and does not contribute to the value of the determinant.