Evaluate each determinant in Exercises 49–52. - 2 - 3 3 51 - 4 0 01 2 2 - 32 0 1 1

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible (a non-zero determinant indicates invertibility) and the volume scaling factor of the linear transformation represented by the matrix. For a 2x2 matrix, the determinant is calculated as ad - bc, while for larger matrices, it can be computed using methods like cofactor expansion or row reduction.

Cofactor expansion is a method used to calculate the determinant of a matrix by breaking it down into smaller matrices. This involves selecting a row or column and multiplying each element by its corresponding cofactor, which is the determinant of the submatrix formed by removing the row and column of that element, adjusted by a sign based on its position. This technique is particularly useful for larger matrices, allowing for a systematic approach to finding the determinant.

Row reduction, or Gaussian elimination, is a process used to simplify a matrix to its row echelon form or reduced row echelon form. This technique involves performing elementary row operations, such as swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting rows. Row reduction is not only useful for solving systems of equations but also for calculating determinants, as it can simplify the matrix to a form where the determinant can be easily computed.