- This page gives an overview of all articles in the 1911 Brittanica which are alphabetized under
**Alk**to Alt. - Recalling the formula A =a11A11+a12Al2+a13A13+ï¿½ï¿½ï¿½+a1nA1n, it will be seen that a ik and
**Alk**involve corresponding determinants. - Since A lk is a determinant we similarly obtain
**Alk**= a21Alk+ï¿½ ï¿½ ï¿½ +a2,k-iAl,k +a2,k+lAl,k+ ï¿½ï¿½ï¿½+a2 21 2,k-1 2, k +1 2,n and thence = Xalia2kAli where k; i,k 2k and as before A = a1, an A i> k i,k I ail, auk 12k an important expansion of A. - If the jth column be identical with the i ll ' the determinant A vanishes identically; hence if j be not equal to i, k, or r, a 11 a 21 a31 0 =I
**alk**a2k a3k A11.

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