GATE CSE 2015 | Set 2 | Question: 27

Question

Perform the following operations on the matrix $\begin{bmatrix} 3 & 4 & 45 \\ 7 & 9 & 105 \\ 13 & 2 & 195 \end{bmatrix}$

1. Add the third row to the second row
2. Subtract the third column from the first column.

The determinant of the resultant matrix is _____.

Comments

Yes @Biswajit Kumar you are correct.
It's a great question to test your conceptual understanding and focus while solving questions.
We all know that Elementry row or column operations on the determinant have no effect on the determinant's value, so we don't need to perform any of the specified operations.
YET! The question is framed so that most of us will perform these operations and get the answer in the long run, wasting time!!! 


P.S.- I also didn't read it properly and calculated the determinant the Long way!!😅

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Only interchanging two rows or two columns determinant value becomes -ve .Except any elementary operation doesn't affect determinant.

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fell on the same hole

Answer 1

Answer $0$, because it is easy to see that first column and third column are multiple of each other.

Third column $=$ First column $^{\ast }15.$

So rank is  $< 3$, so Determinant must be $0$.

It stays zero as row & column transformations don't affect determinant.

Comments

the final matrix is correct and the determinant of the matrix is zero for the final matrix. May be u have done wrong calculations.here is the det of the abv matrix

=>3(11*182-2*280)-4(20*182-13*280)+42(20*2-13*11)

=4326-0-4326

=0

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mine is also coming the same 1st column ko multiply karna pada from 14 then i got last column

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Akash kanase is kobe in clutch time

Answer 2

Ans Zero, row and column transformations doesn't affect determinant.

Comments

pls anyone show the matrix after transformation

Answer 3

Here is the solution : 

Answer 4

Only interchanging two rows or two columns determinant value becomes -ve .Except any elementary operation doesn't affect determinant.

Answer  0

Comments

Rank is 2, which is less than the order(=3)

So need not calculate the determinant.

We can just put 0

Answer 5

If two rows of a matrix are interchanged, the determinant changes sign. If a multiple of a row/column is subtracted from another row/column, the value of the determinant is unchanged.

so Det(A)=0 as after taking common 15  from  column 3  column 1 and column 3 become identical so Det(A)=0.

Answer=0

Answer 6

Basically, if you multiply the first column by 15 and it becomes identical to the third column, the determinant is 0.

In simpler terms:
If one column is just a multiple of another, the columns are "dependent," which always makes the determinant zero.