Question

How do you find the determinant of a 3 x 3 matrix?

Answer 1

@rational

Answer 2

try this
https://www.khanacademy.org/math/precalculus/precalc-matrices/inverting_matrices/v/finding-the-determinant-of-a-3x3-matrix-method-1

Answer 3

\[\begin{vmatrix} a&b&c\\d&e&f\\g&h&i \end{vmatrix}\]

Answer 4

um, what? XD

Answer 5

first ignore first row and first column :

\[\begin{vmatrix} \color{gray}{ a}&\color{gray}{ b}&\color{gray}{ c} \\\color{gray}{d}&e&f\\\color{gray}{g}&h&i \end{vmatrix} \]


the first term of determinant would be \(a(ei-hf)\)

Answer 6

did u watch that video

Answer 7

yes

Answer 8

but I'm confused.

Answer 9

if I had a graphing calculator this would be easy XD

Answer 10

lets work 2-3 quick examples

Answer 11

\[\begin{vmatrix} 1&2&3\\4&5&6\\7&8&9\end{vmatrix} =? \]

Answer 12

there will be 3 terms in the determinant
to get the first term ignore first row and first column, calculate the determinant of remaining 2x2 matrix :



\[\begin{vmatrix} \color{gray}{1}& \color{gray}{2}& \color{gray}{3}\\ \color{gray}{4}&5&6\\ \color{gray}{7}&8&9\end{vmatrix} \]

Answer 13

can you find the determinant of that bottom right 2x2 matrix ?

Answer 14

yes, -3?

Answer 15

Yes! we call "-3" the cofactor of "1"

Answer 16

why?

Answer 17

the first term in the determinant would be \(1(-3) = -3\)

Answer 18

save that and lets find the second term,
is it okay if i answer that question after we find the determinant ?

Answer 19

sure

Answer 20

how do I find the second term?

Answer 21

to get the second term ignore first row and second column, calculate the determinant of remaining 2x2 matrix :



\[\begin{vmatrix} \color{gray}{1}& \color{gray}{2}& \color{gray}{3}\\ 4&\color{gray}{5}&6\\ 7&\color{gray}{8}&9\end{vmatrix} \]

Answer 22

row, being "1 2 & 3" right?

Answer 23

just making sure I didn't get it confused..

Answer 24

yes

Answer 25

rows are horizontal, and columns vertical, right?

Answer 26

we're on same page

Answer 27

-6?

Answer 28

Yes.
so the second term would be \(-2(-6) = 12\)

Answer 29

save that and lets find the third term

Answer 30

let me guess, first row, third column?

Answer 31

and why is it -2?

Answer 32

to get the third term ignore first row and third column, calculate the determinant of remaining 2x2 matrix :



\[\begin{vmatrix} \color{gray}{1}& \color{gray}{2}& \color{gray}{3}\\ 4&5&\color{gray}{6}\\ 7&8&\color{gray}{9}\end{vmatrix} \]

Answer 33

-3

Answer 34

Yes so the third term would be \(3(-3) = -9\)

Answer 35

Add these 3 terms to get the determinant

Answer 36

determinant = \(-3 + 12 - 9 = ?\)

Answer 37

so why was it 1, -2, and 3? why was 2 negative?

Answer 38

and the answer is 0

Answer 39

thats a good question!
the sign is obtained using the formula : \(\large (-1)^{m+n}\)
where m = row number
n = column number

Answer 40

you will be good with determinants if you just remember this :

flip the sign of second term

Answer 41

okay...

Answer 42

so will you check this for me?

Answer 43

sure ask

Answer 44

one sec, still working it out

Answer 45

how do you make a 3 x 3 matrix on here? XD

Answer 46

```
\begin{vmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{vmatrix}
```

produces below
\[\begin{vmatrix} a&b&c\\d&e&f\\g&h&i \end{vmatrix}\]

Answer 47

okay thanks

Answer 48

\(\begin{vmatrix}-4&5&6\\0&4&4\\-2&-5&4\end{vmatrix}\)

Answer 49

i got -20

Answer 50

show steps for the three terms

Answer 51

okay.
1(-4)
-2(-4)
3(-8)
-4+8-24=-20

Answer 52

@rational

Answer 53

NO

Answer 54

oh...

Answer 55

\[\begin{vmatrix}-4&5&6\\0&4&4\\-2&-5&4\end{vmatrix} = -4(16+20) -5(0+8) +6(0+8) = ?\]

Answer 56

isn't it 16+-20? and 0+-8? and, again 0+-8?

Answer 57

and why is it 4, 5, and 6?

Answer 58

determinant of 2x2 matrix is ad-bc right

Answer 59

yes.

Answer 60

whats the determinant of

| 4 4 |
|-5 4 |

?

Answer 61

-4

Answer 62

wait, no

Answer 63

wow, I've been adding this entire time XD

Answer 64

thanks for bringing that to my attention. 16--20=16+20=36

Answer 65

so why is it 4, 5, and 6?

Answer 66

look at first row

Answer 67

ohhh okay.

Answer 68

44?

Answer 69

http://www.wolframalpha.com/input/?i=determinant+%28%28-4%2C5%2C6%29%2C%280%2C4%2C4%29%2C%28-2%2C-5%2C4%29%29

Answer 70

what did I do wrong?...

Answer 71

\[\begin{vmatrix}-4&5&6\\0&4&4\\-2&-5&4\end{vmatrix} = -4(16+20) -5(0+8) +6(0+8) = -4(36) - 40 + 48 = ?\]

Answer 72

I see what I did wrong xp I forgot to multiply -4 and 36 XD

Answer 73

well, I did, but when I did the last step, I totally forgot to put -144 in and put 36

Answer 74

now I got -136 :)

Answer 75

thank you

Answer 76

nice :)