Problem 54 - College Algebra

Question

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

Step-by-Step Solution

Answer

It is necessary to supply minus signs when expanding a determinant by minors whenever the sum of the row number 'i' and column number 'j' (applying \(i + j\)) is an odd number. It is because the term should be subtracted from the determinant value due to the rule of signs in determinants.

Step 1: Understanding Determinants and Minors

Determinants are calculated for square matrices and have important properties in linear algebra. A minor, in linear algebra, is the determinant of a square submatrix. So, when we say 'expanding a determinant by minors', we are basically referring to a method where each element of the matrix is multiplied by its minor and the results are subtracted or added according to a certain pattern to find the determinant value.

Step 2: Rule of Signs in Determinants

When expanding determinants by minors, the sign associated with each term of the expansion follows a checkerboard pattern, or more specifically it is determined by the formula \((-1)^{i+j}\) where \(i\) is the row number and \(j\) is the column number. If the sum of the row number and the column number is even (\(i+j\)), you add the term to the determinant result, but if it's odd, you subtract the term. This reflects into the rule of alternating signs that determinants follow.

Step 3: Application of Minus Signs

Thus, it's necessary to supply the minus signs during the expansion when the sum of the row number and column number is odd. These minus signs are critical in calculating the correct value of the determinant.