## anonymous one year ago **WILL GIVE MEDALS, SUPER EASY QUESTION** https://gyazo.com/93f26f1944e2f6f171354d917159af95

1. jim_thompson5910

based on these properties http://math.kennesaw.edu/~plaval/math3260/det2.pdf if we multiply any column by a constant k, then det(A) = k*det(A)

2. anonymous

okay , how does this help me in this question?

3. anonymous

is the first one 12? I think i got the first one, second is tricky

4. jim_thompson5910

$\det\left[ \begin{array}{ccc} a & 6 & d\\ b & 6 & e\\ c & 6 & f\\ \end{array} \right] = \det\left[ \begin{array}{ccc} a & 6*1 & d\\ b & 6*1 & e\\ c & 6*1 & f\\ \end{array} \right]$ $\det\left[ \begin{array}{ccc} a & 6 & d\\ b & 6 & e\\ c & 6 & f\\ \end{array} \right] = 6*\det\left[ \begin{array}{ccc} a & 1 & d\\ b & 1 & e\\ c & 1 & f\\ \end{array} \right]$ $\det\left[ \begin{array}{ccc} a & 6 & d\\ b & 6 & e\\ c & 6 & f\\ \end{array} \right] = 6*2$ $\det\left[ \begin{array}{ccc} a & 6 & d\\ b & 6 & e\\ c & 6 & f\\ \end{array} \right] = 12$

5. anonymous

yes i got it! can you help me with the second one?

6. anonymous

i can't seem to get the second one for some reason

7. jim_thompson5910

still thinking on that one

8. anonymous

i still can't figure it out lol

9. jim_thompson5910

I think I might have something. One second

10. jim_thompson5910

$\det\left[ \begin{array}{ccc} a & -4 & d\\ b & -7 & e\\ c & -10 & f\\ \end{array} \right] = \det\left[ \begin{array}{ccc} a & -3-1 & d\\ b & -6-1 & e\\ c & -9-1 & f\\ \end{array} \right]$ $\det\left[ \begin{array}{ccc} a & -4 & d\\ b & -7 & e\\ c & -10 & f\\ \end{array} \right] = \det\left[ \begin{array}{ccc} a & -3 & d\\ b & -6 & e\\ c & -9 & f\\ \end{array} \right] - \det\left[ \begin{array}{ccc} a & 1 & d\\ b & 1 & e\\ c & 1 & f\\ \end{array} \right]$ $\det\left[ \begin{array}{ccc} a & -4 & d\\ b & -7 & e\\ c & -10 & f\\ \end{array} \right] = \det\left[ \begin{array}{ccc} a & -3*1 & d\\ b & -3*2 & e\\ c & -3*3 & f\\ \end{array} \right] - \det\left[ \begin{array}{ccc} a & 1 & d\\ b & 1 & e\\ c & 1 & f\\ \end{array} \right]$ $\det\left[ \begin{array}{ccc} a & -4 & d\\ b & -7 & e\\ c & -10 & f\\ \end{array} \right] = -3*\det\left[ \begin{array}{ccc} a & 1 & d\\ b & 2 & e\\ c & 3 & f\\ \end{array} \right] - \det\left[ \begin{array}{ccc} a & 1 & d\\ b & 1 & e\\ c & 1 & f\\ \end{array} \right]$ $\det\left[ \begin{array}{ccc} a & -4 & d\\ b & -7 & e\\ c & -10 & f\\ \end{array} \right] = -3*(-5)-(2)$ $\det\left[ \begin{array}{ccc} a & -4 & d\\ b & -7 & e\\ c & -10 & f\\ \end{array} \right] = 13$

11. anonymous

perfect, thanks man

12. jim_thompson5910

no problem