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in this video we're going to talk about
how to determine the inverse of a 3x3
matrix so let's say if we have matrix a
and it contains the elements 1 2
negative 1 negative 2 0 1 1 negative 1
and 0 so go ahead and determine the
inverse of matrix a so first we need to
rewrite this in the form of an Augmented
matrix with matrix a and the
multiplicative identity of a 3x3 matrix
so is matrix a it's 1 2 negative 1 and
so forth and the multiplicative identity
matrix it's going to be 1 0 0 0 1 0 0 0
1 so now here's what we need to do we
need to perform elementary row
operations and basically take the left
side and make it look like the right
side now the operations that we apply to
the left side we have to apply to the
right side so the right side is going to
change as a result and whatever the
result is on the right side that is
going to be the inverse of matrix a so
let's begin by turning this number into
a 0 to do that let's subtract Row 1 by
Row 3 and let's apply the result to Row
3
now Row 1 is not gonna change so let's
rewrite that Row 2 is also not going to
change as well now for Row 3 it's gonna
be 1 minus 1 so that's 0 and then it's 2
minus negative 1 or 2 plus 1 which is
stream and then we're gonna have
negative 1 minus 0 so that's negative 1
and then 1 minus 0 is 1 0 minus 0 is 0
and then 0 minus 1 is negative 1 so now
what do you think we need to do next how
can we turn this negative 2 into a 0
what row operations do we need to apply
to row 2 to do so so we're going to say
2 times R 1 plus R 2 let's do that so
we're change in Row 2 Row 1 is going to
stay the same Row 3 will remain the same
as well so 2 times 1 plus negative 2
that's going to be 0 2 times 2 plus 0
that's going to be 4 2 times negative 1
which is negative 2 plus 1 that's
negative 1/2 times 1 plus 0 is going to
be 2 2 times 0 plus 1 that's 1 2 times 0
plus 0 is 0 now let's turn that number
into a 0 so we need to apply the row
operation to Row 3 I'm gonna multiply
our 2 by 3 because 3 times 4 is 12 and
then I'm gonna subtract it by 4 times R
3 because negative 4 times 3 is negative
12 nope
so it's a zero so Row one it's gonna
stay the same and let's rewrite Row two
so we're gonna have three times zero
minus 4 times 0 which is 0 and then 3
times 4 minus 4 times 3 that's going to
be 0 as well and then 3 times negative 1
which is negative 3 minus 4 times
negative 1 so we have negative 3 plus 4
that's gonna equal to 1 and then 3 times
2 is 6 minus 4 times 1 so 6 minus 4 is 2
3 times 1 minus 4 times 0 that's 3 & 3
times 0 minus 4 times negative 1 is 4
now what we need to do is to turn this
number into a zero so that's going to be
straightforward we just got to add in
Row 1 and Row 3 to accomplish that
so let's rewrite rows 2 & 3
so it's gonna be 1 plus 0 which is 1 and
then 2 plus 0 that's 2 negative 1 plus 1
is 0 and then 1 plus 2 is 3 and then 0
plus 3 is 3 and 0 plus 4 is 4 now let's
focus on that number what row operations
do we need to apply to Row 2 in order to
make it 0 all we need to do here now is
add Row 2 and Row 3 so notice what's
going to happen so let's rewrite Row 1
and let's rewrite row three
so Row 2 plus Row 3 0 plus 0 is 0 and
then 4 plus 0 is 4 negative 1 plus 1
that's going to be 0 and then we have 2
plus 2 which is 4 and 1 plus 3 that's 4
and then 0 plus 4 is also 4 now we'll
need to do at this point
wait hold on I didn't copy this
correctly
this is supposed to be 1 2 0 & 3 3 4
so everything else should be fine at
this point this is what I should have
right now so at this point we need to
turn this number into a zero so we need
to apply the operation to Row one so
let's take two times R one and subtract
it by our two so we're going to have
zero zero one two three four and then
zero four zero four four four now we're
going to take two times one minus zero
so that's gonna stay one and then two
times two minus 4 that is now 0 2 times
0 minus 0 is 0 and then 2 times 3 which
is 6 minus 4 that's 2 the next one's
gonna be the same 2 times 4 is 8 minus 4
is 4
and one thing I do need to fix two times
one minus zero that's supposed to be two
and have a 1 here so let's fix that now
we need to do is multiply the first row
by 1/2 and the second row by 1 over 4
and that should give us what we need so
half of 2 is 1 and half of 4 is 2 1/4 of
4 is 1 so all of these will be 1 and
then the last one it's not going to
change so notice that here we have the
multiplicative identity matrix I 3 and
this side represents the inverse of
matrix a 1 1 2 1 1 1 2 3 4 now to
confirm that it's indeed the inverse
what we need to do is multiply matrix a
by the inverse of a and show that it's
equal to I 3 so in other words we need
to take matrix a which was 1 2 negative
1 negative 2 0 1 1 negative 1 0 and
multiply it by the inverse of that
matrix which is 1 1 2 and 1 1 1 2 3 4
and if this is the matrix of a if we did
it correctly we should get this answer 1
0 0 0 1 0 0 0 1 so let's find out if we
did it correctly so I'm gonna put my
answers in here first we need to take
the first row and multiply it by the
first column and add the products so
it's going to be 1 times 1 so we're
multiplying these two first and then
it's gonna be plus 2 times 1 that's
these 2
and then plus negative 1 times 2 so 1
times 1 is 1
2 times 1 is 2 negative 1 times 2 is
negative 2 these two cancel and so we
get 1 so we have the first entry now
let's multiply Row one by column two so
it's gonna be 1 times 1 plus 2 times 1
plus negative 1 times 3 so this is 1
plus 2 minus 3 1 plus 2 is 3 3 minus 3
is 0 now let's try room 1 by column 3 so
that's 1 times 2 plus 2 times 1 plus
negative 1 times 4 so this is 2 2 minus
4 2 plus 2 is 4 4 minus 4 is 0 now let's
move on to Row 2 and let's multiply it
by column 1
so it's gonna be negative 2 times 1 plus
0 times 1 plus 1 times 2 so this is
negative 2 plus 0 plus 2 negative 2 plus
2 is 0 now let's take Row 2 and multiply
it by column 2 so it's going to be
negative 2 times 1 plus 0 times 1 plus 1
times 3 so that's negative 2 plus 0 plus
3 negative 2 plus 3 is 1 now let's
multiply Row 2 by column 3 so negative 2
times 2 plus 0 times 1 plus 1 times 4 so
this is gonna be negative 4 plus 0 plus
4 which adds up to 0
now Row three times column one let's try
that so it's going to be 1 times 1 plus
negative 1 times 1 plus 0 times 2 so
it's 1 minus 1 plus 0 which is going to
be 0 next is going to be Row 3 and then
column 2 so this takes some time I mean
it's pretty laborious but that's how you
can confirm it so it's gonna be 1 times
1 plus negative 1 times 1 and then 0
times 3 so this is 1 minus 1 plus 0
which is 0
and finally the last one it's going to
be Row 3 times column 3 so that's going
to be 1 times 2 plus negative 1 times 1
plus 0 times 4 so it's 2 minus 1 plus 0
2 minus 1 is 1 and so we do get the
identity matrix so that tells us that
this is indeed the inverse of a so now
you know how to find the inverse of a
3x3 matrix
thanks for watching
