## sh3lsh one year ago Determinant question

1. sh3lsh

$A =\left| \begin{array}{ccc} v_{1} \\ v_{2} \\ v_{3}\\v_{4}\end{array} \right| \det = 8$ What is the det of $\left| \begin{array}{ccc} 6v_{1} +2v_{4} \\ v_{2} \\ v_{3}\\v_{4} + 3v_{1}\end{array} \right|$

2. terenzreignz

Gotta love row operations ^^ Let's be general here: $\det\left[ \begin{array}{ccc} v_{1} \\ v_{2} \\ v_{3}\\v_{4}\end{array} \right] =k$

3. terenzreignz

Three elementary row operations: switch, multiply, and "add row multiple" All these have distinct effects on the determinant of the original matrix ^^

4. sh3lsh

Yeah! So I thought the answer would be that it would just be scaled by 3, because scalar addition or subtraction doesn't affect the determinant! However, the answer says it's 0, because the first row is twice the last. That didn't make much sense to me!

5. terenzreignz

When you switch one pair of rows, such as here: $\det\left[ \begin{array}{ccc} v_{1} \\ \color{red}{v_{3}} \\ \color{blue}{v_{2}}\\v_{4}\end{array} \right] =-k$ the determinant gets negated. And will be negated for every pair of rows that you switch :D

6. terenzreignz

The first row is indeed twice the last. Are you finding it hard to believe that that causes the determinant to be zero? ^^

7. sh3lsh

Oops, scaled by 6! (v1 in first row is scaled by 6) Yeah, how does that change anything if the 4th row is a multiple of the first? Does that mean that the 4th row is redundant and if I rref'ed it the row would be 0, so if I tried to take the diagonals it would be 0?

8. terenzreignz

Okay. Let's take a weird approach to this. First, what I said: Switching a pair of rows causes the determinant to be negated or multiplied by a factor of (-1). Next, multiplying a row by a scalar in turn scales the determinant by that scalar. ie $\det\left[ \begin{array}{ccc} v_{1} \\ \color{green}{n}v_{2} \\ v_{3}\\v_{4}\end{array} \right] =\color{green}{n}k$

9. terenzreignz

So far, so good, yes? ^^

10. terenzreignz

So suppose we have a matrix where the first row is a multiple of the last row, and suppose its determinant was d. Like this $\Large \det\left[ \begin{array}{ccc} mu_{3} \\ u_{1} \\ u_{2}\\u_{3}\end{array} \right]=h$

11. terenzreignz

If we switch the first and fourth rows, as per the rule, the determinant gets multiplied by (-1) We get $\Large \det\left[ \begin{array}{ccc} \color{red}{u_{3}} \\ u_{1} \\ u_{2}\\ \color{red}{mu_{3}}\end{array} \right]=\color{red}{-h}$

12. terenzreignz

If we scale the first row by m, as per the other rule, the determinant gets scaled as well by a factor of m. $\Large \det\left[ \begin{array}{ccc} \color{red}mu_{3} \\ u_{1} \\ u_{2}\\mu_{3}\end{array} \right]=-h\color{red}{(m)}$

13. terenzreignz

Now we scale the fourth row by 1/m. Again, by the rules, this would also scale the determinant by a factor of 1/m. $\Large \det\left[ \begin{array}{ccc} mu_{3} \\ u_{1} \\ u_{2}\\ \color{red}{\left(\frac1m\right)}mu_{3}\end{array} \right]=-h(m)\color{red}{\left(\frac1m\right)}$

14. terenzreignz

Simplifying, we get $\Large \det\left[ \begin{array}{ccc} mu_{3} \\ u_{1} \\ u_{2}\\u_{3}\end{array} \right]=-h$ We ended up with our original matrix, but THIS time, we have shown that its determinant is also equal to -h. That can only mean h = -h Or h = 0 Done ^^

15. sh3lsh

You are awesome! Thanks!

16. terenzreignz

No problem ^^