in Linear Algebra
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35 votes
35 votes

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 _____.

in Linear Algebra
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2 Comments

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One keynote is GATE never asks dumb questions and we aren't expected to solve for determinant manually. Approach should be applying properties and get ans in one go.
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4 Answers

45 votes
45 votes
Best answer
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.
edited by

4 Comments

edited by

Third column = First column ∗15.

Final matrix would be --

 3     4      42

20   11   280

13    2     182

It’s 14. Someone check it...

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@ASNR1010

Adding or Subtracting Row/Col doesn’t change determinant.

In initial matrix and in transformed matrix , Determinant will be same. After operations, matrix will change but Determinant will not.

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

1 comment

pls anyone show the matrix after transformation
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1 vote
1 vote
Only interchanging two rows or two columns determinant value becomes -ve .Except any elementary operation doesn't affect determinant.

Answer  0

1 vote
1 vote

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:

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