Linear Algebra

302 Chapter Four. DeterminantsThe second sentence follows from property (3). Multiply the zero row bytwo. That doubles the determinant but it also leaves the row unchanged andhence leaves the determinant unchanged. Thus the determinant must be zero.For the third sentence, where T → · · · → ˆT is the Gauss-Jordan reduction,by the definition the determinant of T is zero if and only if the determinant of ˆTis zero (although the two could differ in sign or magnitude). A nonsingular TGauss-Jordan reduces to an identity matrix and so has a nonzero determinant.A singular T reduces to a ˆT with a zero row; by the second sentence of thislemma its determinant is zero.The fourth sentence has two cases. If the echelon form matrix is singularthen it has a zero row. Thus it has a zero on its diagonal, so the product downits diagonal is zero. By the third sentence the determinant is zero and thereforethis matrix’s determinant equals the product down its diagonal.If the echelon form matrix is nonsingular then none of its diagonal entries iszero so we can use property (3) to get 1’s on the diagonal (again, the verticalbars | · · · | indicate the determinant operation).∣ t 1,1 t 1,2 t ∣∣∣∣∣∣∣∣∣ 1,n1 t 1,2 /t 1,1 t 1,n /t 1,10 t 2,2 t 2,n0 1 t 2,n /t 2,2. = t ..1,1 · t 2,2 · · · t n,n ·. ..∣ 0 t n,n∣0 1 ∣Then the Jordan half of Gauss-Jordan elimination, using property (1) of thedefinition, leaves the identity matrix.1 0 00 1 0= t 1,1 · t 2,2 · · · t n,n ·. ..= t 1,1 · t 2,2 · · · t n,n · 1∣0 1∣So in this case also, the determinant is the product down the diagonal.QEDThat gives us a way to compute the value of a determinant function on amatrix: do Gaussian reduction, keeping track of any changes of sign caused byrow swaps and any scalars that we factor out, and finish by multiplying downthe diagonal of the echelon form result. This algorithm is just as fast as Gauss’sMethod and so practical on all of the matrices that we will see.2.5 Example Doing 2×2 determinants with Gauss’s Method∣ 2 4∣∣∣∣ ∣−1 3∣ = 2 40 5∣ = 10doesn’t give a big time savings because the 2×2 determinant formula is easy.However, a 3×3 determinant is often easier to calculate with Gauss’s Method