Question

Compare the various methods of calculating a determinant of a matrix. Which method requires the least amount of computation?

Answer 1

There are mainly 2 ways for calculating the values of determinants.

I Use of Minors and co-factors:

The minor M_ij of the entry a_ij in a nxn matrix A is the determinant of the matrix obtained by eliminating the ith row and the jth column of A. The cofactor, C_ij is defined by C_ij =(-1)^i+j M_ij.

For a given square matrix he determinant det(A) or, |A| is obtained by cofactor expansion as under:

1. First, select a row or a column of A containing the most zeros.

2.Multiply each entry a_ij of the row/column selected by its corresponding cofactor C_ij.

3.Add these results.

II. Use of Row Operations:

This method is very useful in calculating the values of large determinants.

Various rules regarding the use of this method are as under:

1. The interchanging of two rows or columns of a determinant changes the sign of the determinant.
2. Adding a multiple of one row to another does not have any effect on the value of the determinant.
3. Multiplying a row of a determinant by a constant, scales up the value of the determinant by that constant.
4. The determinant of a lower triangular matrix/ an upper triangular matrix is the product of the diagonal entries.

The second method requires the least amount of computation in case of large determinants.