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Unformatted text preview: ltiple of another row moving from one matrix to the next. Furthermore, since
C is upper triangular, det(C ) is the product of the entries on the main diagonal, in this case
det(C ) = (3)(−1) 133 = −13. This demonstrates the utility of using row operations to assist in
calculating determinants. This also sheds some light on the connection between a determinant and
invertibility. Recall from Section 8.4 that in order to ﬁnd A−1 , we attempt to transform A to In
using row operations
A In Gauss Jordan Elimination −− − − − − − −→
−−−−−−−− I n A− 1 As we apply our allowable row operations on A to put it into reduced row echelon form, the
determinant of the intermediate matrices can vary from the determinant of A by at most a nonzero
multiple. This means that if det(A) = 0, then the determinant of A’s reduced row echelon form
must also be nonzero, which, according to Deﬁnition 8.4 means that all the main diagonal entries
on A’s reduced row echelon form mus...
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