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okay and this video I want to talk about
finding the determinant of a 3x3 matrix
so here's gonna be my matrix a and again
the entries are a sub 1 1 1 2 1 3 2 1 2
2 2 3 3 1 3 2 3 3 I'm kind of just again
a generic way of writing a matrix and
the notation is you know for the
determinate of a you either write Det of
a or they put it looks like absolute
value and it says to compute the
determinant of a 3x3 matrix basically
what you do is if you look at stuff
along the top okay so I'm gonna look at
my a 1 1 basically the formula is you
take that term a11 and I imagine kind of
a kind of cancelling out that row or
excuse me that column and that row I
think what am I left with I'm left with
a smaller matrix a.22 a23 a32 a33 okay
so I've got to take this coefficient a11
times this smaller determinant and
remember to calculate a smaller
determinant you multiply the diagonals
and then subtract the product of the
other kind of diagonal so I assume if
you if you're looking at a 3 by 3 matrix
you know how to do a 2 by 2 but we'll
talk about this in a second as well just
in case so now we do the kind of the
same thing we look at the same same row
but we move over ok we look at that guy
it turns out that there ends up you have
to change the sign so it's going to be a
sub 1 2 and again I think if I were to
cover up that column and that row the
stuff that I would be left with would be
that a 2 1 a 2 3 the a sub 3 1 and the a
sub 3 3
okay looks like I'm running out of room
here and then we go to the last one as
well so okay so the same thing I look at
the a sub 1 3
I imagine covering up that row and that
column and I'm left with the smaller
determinant to compute a 2 1 a 2 2 a 3 1
and a 3 2 okay so this is kind of the
the basic formula for computing a 3 by 3
determinant so pretty tedious but let's
see if we can't can't do one here so
let's let's come up with some specific
numbers I don't know I'm just going to
make up a matrix at random hopefully the
numbers won't be too terrible so I don't
know how about 1 6 4 I'm gonna keep
everything positive just hopefully to
keep keep it keep it straightforward one
six four two seven three eight nine what
are we missing five about five okay so
let's compute the determinant of this
matrix so again it says the determinant
the Det of that matrix a it says again
we look at the the first value which is
a 1 and again if I were to cover up that
row and column column in row
I've got to compute the smaller the
determinant of the smaller matrix 7 3 9
5 again it says we put a minus sign in
the middle ok if I look at the next
entry it's gonna be a 6 so that's what
I'm gonna put in there next and again if
I cover up that column with the 6 and
that row what am I left with it looks
like I'm gonna be left with the 2
the 3 the 8 and the 5 and then to that I
have to add on ok so now we just move
over my next value will be a 4 so if I
cover up that column and that row I'm
gonna have 2 7 8 9 left over okay so
that's kind of the set up so now all we
have to do is compute the individual
determinants so remember to compute a
determinant of a 2 by 2 matrix we
multiply the top left and the bottom
right so we'll get 7 times 5 we put a
minus sign in between there and then we
multiply the other diagonals so 3 times
9 then we'll get minus 6 we'll have to
take 2 times 5 minus 3 times 8 and then
plus 4 we'll have to take 2 times 9
minus 8 times 7 okay so now we're almost
there it's just a little bit of
computation at this point so we get 1
times let's see it looks like we get 35
minus 27 so 35 minus 27 let's go ahead
and write it out minus 6 2 times 5 is 10
minus 24 plus 4 times 2 times 9 is 18
minus 56 so 35 minus 27 is going to be 8
times 110 times
excuse me 10 minus 24 will be negative
14 so we'll get negative 6 times
negative 14 let's see so 18 minus 56
what is that negative looks like 38 to
me
so if we simplify this we'll get eight
two negatives make it positive let's see
what is that 84 six times 14 should be
84 and then we'll get a negative four
times 38 so what's that one 52 it looks
like okay you can check my arithmetic
here hopefully I'm not doing any crazy
arithmetic so eight plus 84 that's gonna
give us 92 minus 152 so the determinant
of this would be negative 60 okay
assuming all my arithmetic is correct
so again I think the main thing you know
it's just knowing the setup again just
take the first row first column and then
again you're just basically breaking it
down to smaller determinants again make
sure that you notice that the again the
signs are positive its negative and then
positive it's easy to leave out I think
that minus sign and then things don't
quite work out so all right I hope this
video makes some sense and helps out
feel free to post comments and questions
if you have them hopefully me or
somebody else can point you in the right
direction
