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Again, the determinant is calculated by rewriting it in terms of the elements of the first row mVt times determinants of smaller matrices Mlf, each of which is obtained by eliminating the first row and /th column. This procedure seems straightforward until you realize that the determinant of a 10 x 10 matrix is a function of the determinants of ten 9x9 matrices, which are each in turn functions of determinants of nine 8x8 matrices, etc. In practice, what this means is that we would almost always use a computer to calculate the determinant of a large matrix.

If a matrix contains a row or a column with several zeros in it, it is easier to calculate the determinant by moving across the row with the most zeros (or moving down the column with the most zeros). The only caveat is that we must multiply the determinant by -1 if we use an even-numbered row or an even-numbered column. Thus, we can generalize Rule P2.17 to allow us to use the kth row to find the determinant:

Rule P2.18a:

|M| = (-1 )*+1 2 (-1 y+1 mkj |Mkj| (determinant of a d x d matrix)

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