80 a31 a32 a 1 a 2 a33 a32 a 13 a22 a 1 a23 a 1 a32 a3 a A a 1 a 12 a33 a21 932 913 a23 a31 a12 a11 a22 a33 913 931 922 Expansion along first Column C By expanding along C we get A a 1 1 922 MATHEMATICS a11 912 a13 A a21 922 923 a31 a32 A33 iii Let A 923 a32 a33 t to me jepub 2 2 23 4 0 a12 a21 a33 a12 923 a31 and B 921 1 1 Rationalised 2023 24 1 20 a31 a 13 a 12 a13 931 1 1 922 923 a 1 a 2 a33 a23 a32 a21 a 12 a 33 a 3 a32 a31 a 12 a23 a13 a22 13 a22 a31 a 13 A a a22 a33 a 11 a23 a32 a21 a 12 a33 a31 a 13 a22 a 1 a 22 a33 a 11 a 23 a32 a 12 a21 A33 A 2 A23 A31 a 13 a21 a32 12 A 0 8 8 and B 0 2 2 Observe that A 4 2 2 B or A square matrices A and B 912 a13 a32 a33 1 24 3 0 4 1 0 a23 a11 a32 a13 a31922 3 Clearly values of A in 1 2 and 3 are equal It is left as an exercise to the reader to verify that the values of A by expanding along R C and C are equal to the value of A obtained in 1 2 or 3 Hence expanding a determinant along any row or column gives same value Remarks i For easier calculations we shall expand the determinant along that row or column which contains maximum number of zeros Example 3 Evaluate the determinant A 1 a13 a21 a32 a21 a 13 a32 ii While expanding instead of multiplying by 1 we can multiply 1 or 1 according as i j is even or odd 2 a31 a 12 a23 verify that A 2B Also In general if A kB where A and B are square matrices of order n then A k B where n 1 2 3 where n 2 is the order of