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I am currently studying for a test, and I understand how to solve determinants when given concrete values. However, I have no idea how to solve this practice problem. Can someone lead me in the right direction to the solution?

Given

$det\begin{bmatrix}a & b & ... & c\\d & e & ... & f\\... & ... & ... & ...\\g & h & ... & i\end{bmatrix} = K$

Find the $det\begin{bmatrix}a & b & ... & c\\3d+a & 3e+b & ... & 3f+c\\... & ... & ... & ...\\g-d & h-e & ... & i-f\end{bmatrix}$

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  • $\begingroup$ Are the rows in the middle changed at all? $\endgroup$ – EuYu Oct 17 '12 at 2:12
  • $\begingroup$ what is the size of the matrix? $\endgroup$ – ILoveMath Oct 17 '12 at 2:16
  • $\begingroup$ The size isn't given. $\endgroup$ – Grace C Oct 17 '12 at 2:48
  • $\begingroup$ I'm not sure about the rows in the middle. That's all that is given. $\endgroup$ – Grace C Oct 17 '12 at 2:48
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From the question, I'm going to make the assumption that the second and last rows are the only ones altered (as I can see no other obvious pattern).

It looks like the new matrix is obtained from the original by

  1. Subtracting row $2$ from the last row
  2. Multiplying row $2$ by $3$
  3. Adding row $1$ to row $2$

How do these elementary row operations affect the determinant?

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  • $\begingroup$ It would certainly alter the k value, but I'm not sure how to denote this. $\endgroup$ – Grace C Oct 17 '12 at 3:58
  • $\begingroup$ Well, from what you learned about determinants, how does each elementary row operation alter the determinant? Or perhaps the better question, how were you taught to evaluate determinants? $\endgroup$ – EuYu Oct 17 '12 at 3:59

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