Property 5
If some or all elements of a row or column of a determinant are expressed as sum of two ( or more ) terms , then the determinant can be expressed as sum of two ( or more ) determinants .
Property 6
If , to each element of any row or column of a determinant , the equi multiples of corresponding elements of other row ( or column ) are added , then value of determinant remains the same , i . e ., the value of determinant remain same if we apply the operation Ri → Ri + kRj or Ci → Ci + k Cj .
o
o
If Δ1 is the determinant obtained by applying Ri → kRi or Ci → kCi to the determinant Δ , then Δ1 = kΔ . If more than one operation like Ri→ Ri + kRj is done in one step , care should be taken to see that a row that is affected in one operation should not be used in another operation . A similar remark applies to column operations .
Numerical : Solve without expanding
Solution : Observe that 65 = 9 x 7 + 2 75 = 9 x 8 + 3