Evaluate each determinant by expanding by cofactors.

Key Concepts

Sign Factor in Cofactors

The sign factor in cofactors accounts for the alternating signs in the cofactor expansion process. For an element located in the i-th row and j-th column, the corresponding sign factor is given by (-1)^(i+j). This alternating pattern ensures that the contributions of different minors are correctly signed, which is crucial for the accurate computation of the determinant.

Minor

A minor is the determinant of a smaller square matrix that is obtained by deleting one or more rows and columns from a larger matrix. In the context of computing cofactors, the minor associated with an element is the determinant of the submatrix formed when the row and column containing that element are removed. The concept of minors is fundamental for the calculation of cofactors and, consequently, for the determinant of larger matrices.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It encapsulates important properties of the matrix, such as whether the matrix is invertible and the scaling factor for linear transformations represented by the matrix. The determinant is also used in solving systems of linear equations and in theoretical aspects of linear algebra, including eigenvalues and eigenvectors.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method for computing the determinant of a matrix. This method involves expanding the determinant along a row or a column, where each term is the product of an element and its corresponding cofactor. The cofactor is calculated as the determinant of the submatrix that remains after removing the element's row and column, multiplied by a sign factor determined by the element's position.

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