Linear Algebra

300 Chapter Four. DeterminantsWe are left with four determinants, such that in each row of each matrix thereis a single entry from the original matrix.3.5 Example In the same way, a 3×3 determinant separates into a sum ofmany simpler determinants. We start by splitting along the first row, producingthree determinants (the zero in the 1, 3 position is underlined to set it off visuallyfrom the zeroes that appear in the splitting).∣ ∣ ∣ 2 1 −1∣∣∣∣∣ 2 0 0∣∣∣∣∣ 4 3 0∣2 1 5 ∣ = 0 1 0∣∣∣∣∣4 3 02 1 5∣ + 0 0 −14 3 02 1 5∣ + 4 3 02 1 5 ∣Each of these three will itself split in three along the second row. Each ofthe resulting nine splits in three along the third row, resulting in twenty sevendeterminants∣ ∣ ∣ ∣ 2 0 0∣∣∣∣∣ 2 0 0∣∣∣∣∣=4 0 0∣2 0 0∣ + 2 0 0∣∣∣∣∣4 0 00 1 0∣ + 2 0 0∣∣∣∣∣4 0 00 0 5∣ + 0 0 −10 3 02 0 0∣ + · · · + 0 0 00 0 5 ∣such that each row contains a single entry from the starting matrix.So an n×n determinant expands into a sum of n n determinants where eachrow of each summands contains a single entry from the starting matrix. However,many of these summand determinants are zero.3.6 Example In each of these three matrices from the above expansion, twoof the rows have their entry from the starting matrix in the same column, e.g.,in the first matrix, the 2 and the 4 both come from the first column.2 0 00 0 −10 1 04 0 00 3 00 0 0∣0 1 0∣∣0 0 5 ∣ ∣0 0 5∣Any such matrix is singular, because in each, one row is a multiple of the other(or is a zero row). Thus, any such determinant is zero, by Lemma 2.3.Therefore, the above expansion of the 3×3 determinant into the sum of thetwenty seven determinants simplifies to the sum of these six.∣ ∣ 2 1 −1∣∣∣∣∣ 2 0 0∣∣∣∣∣ 4 3 0∣2 1 5 ∣ = 2 0 00 3 00 0 5∣ + 0 0 00 1 0∣∣ 0 1 0∣∣∣∣∣ 0 1 0+4 0 0∣0 0 5∣ + 0 0 02 0 0∣∣ 0 0 −1∣∣∣∣∣ 0 0 −1+4 0 0∣0 1 0 ∣ + 0 3 02 0 0 ∣