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My homework question says given:

$$\left|\begin{matrix}a & b& c\\ d& e& f\\ g& h& i\end{matrix}\right| = -6$$

evaluate the determinant

$$\left|\begin{matrix}a+d & b+e& c+f\\ -d& -e& -f\\ g& h& i\end{matrix}\right|$$

It says I have to do this by row reduction. I know how to row reduce but I don't know where to start for this question. I just can't grasp how the two matrixes/determinants can be used together to solve the question.

Any help would be greatly appricated

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Are you sure it says "row reduction" rather than "row operations" or something like that? – Henning Makholm Oct 31 '11 at 18:53
up vote 1 down vote accepted

Have you learned how row operations affect the determinant?
For example,

  1. Interchanging two rows negates the determinant.
  2. Multiplication of one row by a constant multiplies the determinant by that constant.
  3. Adding a multiple of one row to another row does not affect the determinant.

Those rules are what I would use to solve the problem.

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Yes, I did learn these rules. So basically only two things have been done to the matrix? firstly #3 happens, and that doesnt affect the determinant, and then we divide row 2 by the constant -1. So the only thing that changes is that -6 turns into --6. So the determinant would just be 6? Am I doing that right? – Cheesegraterr Oct 31 '11 at 18:54
That's correct. – AMPerrine Oct 31 '11 at 18:55

Hint: what row operation will change the first row from $[a,b,c]$ to $[a+d,b+e,c+f]$? How does that affect the determinant? Then what row operation changes the second row from $[d,e,f]$ to $[-d,-e,-f]$?

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