Evaluate each determinant by expanding by cofactors.

Key Concepts

Determinant

The determinant is a scalar value derived from a square matrix that encapsulates certain properties of the matrix, such as invertibility and the scaling factor of the linear transformation described by the matrix. It plays a crucial role in systems of linear equations, eigenvalue problems, and in evaluating areas or volumes in geometry.

Cofactor Expansion

Cofactor expansion (also known as Laplace expansion) is a method for calculating the determinant of a matrix. This method involves breaking down the determinant into smaller, more manageable determinants by selecting a row or a column, calculating the minors for each element, applying the appropriate signs based on their position, and summing the products. This approach is recursive and works well for both theoretical proofs and practical computations.

Minor

The minor of an element in a matrix is defined as the determinant of the smaller matrix obtained by removing the row and column containing that element. Minors are integral to the process of cofactor expansion, as each minor is used to calculate the cofactor for the corresponding element in the original matrix.

Sign Factor in Cofactor Expansion

The sign factor in cofactor expansion refers to the (-1)^(i+j) term associated with each element a_ij of the matrix, where i is the row index and j is the column index. This alternating sign pattern is essential for accurately computing the determinant using cofactor expansion, ensuring that the contributions of each minor are correctly adjusted based on their positions within the matrix.

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