Final answer:
To compute the determinant, simplify the matrix via row reductions to triangular form and then use cofactor expansion on a row or column with zeros. Row operations can affect the determinant's value, so consider their effects when simplifying the matrix.
Step-by-step explanation:
To compute the determinant of a matrix using a combination of row reduction and cofactor expansion, follow these steps:
- First, apply row reductions to simplify the matrix, aiming for a triangular form which makes the determinant easier to find. This is because the determinant of a triangular matrix is the product of its diagonal elements.
- Next, choose a row or column (preferably one with zeros) to perform cofactor expansion. This involves creating minors of the matrix and then using these minors and their corresponding cofactors to calculate the determinant.
Remember, while row operations like swapping rows reverse the sign of the determinant, multiplying a row by a scalar does not change the determinant's absolute value (but scales it by the scalar), and adding a multiple of one row to another doesn't change the determinant at all. These properties can be very helpful when simplifying the matrix before using cofactor expansion.