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Unformatted text preview: ing determinants and we present that here. Before we state the theorem, we need some
more terminology. 510 Systems of Equations and Matrices Definition 8.14. Let A be an n × n matrix and Aij be deﬁned as in Deﬁnition 8.12. The ij
minor of A, denoted Mij is deﬁned by Mij = det (Aij ). The ij cofactor of A, denoted Cij is
deﬁned by Cij = (−1)i+j Mij = (−1)i+j det (Aij ).
We note that in Deﬁnition 8.13, the sum
a11 det (A11 ) − a12 det (A12 ) + − . . . + (−1)1+n a1n det (A1n )
can be rewritten as
a11 (−1)1+1 det (A11 ) + a12 (−1)1+2 det (A12 ) + . . . + a1n (−1)1+n det (A1n )
which, in the language of cofactors is
a11 C11 + a12 C12 + . . . + a1n C1n
We are now ready to state our main theorem concerning determinants.
Theorem 8.7. Properties of the Determinant: Let A = [aij ]n×n .
• We may ﬁnd the determinant by expanding along any row. That is, for any 1 ≤ k ≤ n,
det(A) = ak1 Ck1 + ak2 Ck2 + . . . + akn Ckn
• If A is the matrix obtained from A by:
– interchanging any two rows, then det(A ) = − det(A).
– replacing a row with a nonzero multiple (say c) of itself, then det(A ) = c det(A)
– replacing a row with itself plus a multiple of another...
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