 Hi and welcome to the session on NG Picker here. Let's discuss the question using the property of determinants And without expanding prove that the determinant 2, 7, 65, 3, 8, 75, 5, 9, 86 is equal to 0 Let's start the solution That delta is equal to our given determinant that is 2, 7, 65, 3, 8, 75, 5, 9, 86 We have to prove that delta is equal to 0 by using the property of determinants Again we can rewrite there as delta is equal to 2, 3, 5 Now second column 2 plus 5, 3 plus 5 and 5 plus 4 C3 is as it is 65, 75, 86 Now we can express delta as a sum of 2 determinants that is 2, 3, 5 2, 3, 5, 65, 75, 86 Plus 2, 3, 5, 5, 4 65, 75, 86 As we have the property that if some or all elements of a row or column of a determinant are expressed as sum of 2 or more terms then the determinant can be expressed as sum of 2 or more determinants That is why we have expressed delta is sum of 2 determinants because we have expressed C2 as a sum of 2 numbers So delta can be expressed as a sum of 2 determinants Now this determinant is also equal to 0 because C1 and C2 are identical We have a property of a determinant if any 2 rows or column of a determinant are identical that is all corresponding elements are same then value of the determinant is 0 Therefore we have delta is equal to 0 plus 2, 5, 65, 3, 5, 75, 5, 4, 86 As C1 and C2 are identical R1 goes to R1 minus R2 to delta Now we get delta is equal to this is 2 minus 3 minus 1, 5 minus 5, 0, 65 minus 75 this is minus 10 Now R2 and R3 are same 5, 4, 86 Again by applying R2 goes to R2 plus 3 R1 and R3 goes to R3 plus 5 R1 we get delta is equal to minus 1, 0, minus 10 Now R2 goes to R2 plus 3 R1 that is 3 plus minus 3, 0, 5 this is 45 Again R3 goes to R3 plus 5 R1 we get 0, 4, 36 Now expanding along first column we get delta is equal to minus 1 into determinant 5, 4, 45, 36 minus 0 into minus 0 plus 0 which is equal to minus 1 into 180 minus 180 minus 0 plus 0 which is again equal to 0 Hence we have proved that our given determinant that is delta is equal to 0 hence proved I hope the question is clear to you by and have a good day