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**Unformatted text preview: **ng replaced is the not the ﬁrst row of A. By deﬁnition,
n det(A ) = a1p C1p ,
p=1 where in this case, a1p = a1p , since the ﬁrst rows of A and A are the same. The matrices
A1p and A1p , on the other hand, are diﬀerent but in a very predictable way − the row in A1p
which corresponds to the row cR in A is exactly c times the row in A1p which corresponds to
the row R in A. In other words, A1p and A1p are k × k matrices which satisfy the induction
hypothesis. Hence, we know det A1p = c det (A1p ) and C1p = c C1p . We get
n n a1p C1p = det(A ) =
p=1 n a1p c C1p = c
p=1 a1p C1p = c det(A),
p=1 which establishes P (k + 1) to be true. Hence by induction, we have shown that the result
holds in this case for n ≥ 1 and we are done.
While we have used the Principle of Mathematical Induction to prove some of the formulas we have
merely motivated in the text, our main use of this result comes in Section 9.4 to prove the celebrated
Binomial Theorem. The ardent Mathematics student will no doub...

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