## anonymous one year ago Evaluate the determinant of the matrix [8. 3 -2] [-1 3 2] [ 5 -1 1]

1. DanJS

scalar triple product

2. anonymous

what?

3. UsukiDoll

cofactor expansion... that's needed to shrink the matrix down to a 2 x 2

4. DanJS

A * (B x C)

5. anonymous

How?

6. DanJS

do you know a 2 by 2 determinant?

7. anonymous

yes! i just don't know how to find the determinant for a 3x3 matrix

8. DanJS

Determinant of 2 by 2 matrix = a*d - cd [a b] [c d]

9. DanJS

ok...

10. UsukiDoll

ad - bc for determinant of a 2 x 2 matrix for a 3 x 3 matrix we have to use cofactor expansion.

11. anonymous

how do i do cofactor expansion?

12. DanJS

|dw:1438052983671:dw|

13. UsukiDoll

Cofactor expansion expanding a row ... find the minors and the cofactors of the first row entries ( you can choose second row or third row entries but the sign pattern is different).

14. DanJS

|dw:1438053078307:dw|

15. DanJS

Cover first column, 8*det of that 2x2 is the first of 3 terms

16. DanJS

look up determinant of 3 x 3 matrix.. lol this is alot of drawing

17. UsukiDoll

the signs for the 3 x 3 matrix is like a checkerboard |dw:1438053171537:dw| that's why I use the first drawing a lot.

18. UsukiDoll

* first row signs

19. DanJS

yeah i just remember the second is minus

20. UsukiDoll

|dw:1438053270643:dw|

21. UsukiDoll

cut row one cut column one cut row one cut column two cut row one cut column three

22. UsukiDoll

|dw:1438053376916:dw|

23. UsukiDoll

ok I'm going to clean this up a bit.. but where I circled the row and column... you draw a line to cross them out... you should be left with 4 numbers... that's your 2 x 2 matrix. now you can take the determinant of those matrices...

24. UsukiDoll

|dw:1438053458361:dw|

25. UsukiDoll

|dw:1438053534643:dw|

26. UsukiDoll

|dw:1438053599579:dw| normally we would have a + sign but that negative -2 on the first row third column entry changes that.

27. UsukiDoll

|dw:1438053728550:dw| so we should have this equation after we have done cofactor expansion. Now we can take the determinant of the 2 x 2 matrix and combine like terms. The final answer for the determinant of the 3 x 3 matrix should be just a single number