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.
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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