Algebra of Matrices - Matrices and Determinants, Business Mathematics & Statistics Video Lecture | Business Mathematics and Statistics - B Com

this is a tutorial on inverse matrices
now this concept is very useful when
working with matrix equations but before
we actually get into the equations let's
find out how to find the inverse of a
matrix so let's look at example 1 so
here we have a matrix and we want to
find its inverse now let's consider a
formula that could help us find the
inverse so say we just had a general
matrix with a b c and d as the values
for its various elements now if we
wanted to find the inverse of that
particular matrix we would need to do
this operation here so the 1 over ad
minus CB is pretty much telling us 1
over the determinant of the given matrix
now we want to take that value and
multiply it by this second matrix here
now if you notice it's just a
manipulation of our original matrix our
a and D elements are going to switch
places and then our B and C elements are
just going to take opposite signs of
from what they currently have so let's
take this formula and apply it to our
matrix from example 1 so we'll put in
the values for a b c and d so we'll have
negative 2 negative 1 6 and 2 now at
this point first thing we want to do is
find the determinant for our matrix and
it's very important to check that first
because if the determinant for this
matrix is 0 then this particular matrix
actually would not have an inverse and
the reasoning behind that is this part
right here the determinant is what's on
the bottom and if that's 0 we would have
1 over 0 being multiplied to this matrix
well 1 over 0 is an undefined number now
because of that
there would be no inverse for any matrix
as a determinant of 0 so let's find the
determinant for our matrix so we'll
first take the negative 2 and multiply
it by a positive 2 and then we'll
subtract the product between 6 and
negative 1 now when we simplify this we
get negative 2 times 2 which is a
negative 4 and then minus a 6 times a
negative 1 which gives us a value of
negative 6 now from here when we further
simplify this expression we have
negative 4 minus a negative 6 well minus
a negative it's just like adding a
positive so we have negative 4 plus 6
giving us a value of 2 for our
determinant so now that we know what the
determinant is we could plug it back in
to our equation right here where it goes
so let's do that now from this point we
need to make this new matrix right here
so we need to switch our a and D values
so our negative 2 and positive 2 we're
going to switch spots to look like this
so now our positive 2 is in the top-left
corner and our negative 2 is in the
bottom right corner now as for our B and
C elements for our matrix you notice I'm
going to take the opposite of each of
them so our negative 1 will now be a
positive 1 and our positive 6 will be a
negative 6 so now from here all we need
to do is take this scaler of 1/2 and
multiply it to every element in this
matrix and then we'll get the inverse
matrix as our result so let's do that so
first of all we have 1/2 times 2 which
gives us a value of 1 and then 1/2 times
1 gives us a value of 1/2
and then 1/2 times the negative 6 gives
us negative 3 and then 1/2 times a
negative 2 gives us a value of negative
1 so for us this matrix right here would
be the inverse of the original matrix we
started with now quick wave that we
could check our answer is to use a
calculator now if you're using a ti-83
plus or 84 plus calculator you can
actually go to the matrix function in
that calculator now to get there you
could push second and then this X to the
negative 1 button that'll take you to
some matrix options now when you're
there you'll see a list of different
matrices labeled a B C D and all the way
through J now what you want to do is go
over to the Edit tab and put in the
elements or values within your matrix
into one of these letters so say for
example we chose matrix a we would go
over to the Edit tab push enter on a and
then put in the corresponding values to
each of our elements for the matrix now
once we've done that and exited we could
go back into the matrix options and
under the tab names we would push enter
on the letter that we put our matrix in
so since we put it into matrix a we'd go
over to a and hit enter and then we'd
see it on our screen now if we want to
get the inverse of that matrix we just
need to push the inverse button which is
this X to the negative 1 button now
alternatively you could just take your
matrix a and then use the caret symbol
to raise it to the negative 1 power in
either case the screen of your
calculator should have something that
looks similar to this matrix a to the
negative 1 now at this point once you
push enter you should get a result that
would show you the inverse of that
particular matrix
now when you do that your answer will
look somewhat similar to this which is
exactly what we got when we calculated
for the inverse of the matrix by hand
now just so you know this doesn't only
work for two-by-two matrices you could
also use this method to find the inverse
of a 3x3 matrix or 4x4 matrix or so
forth now if you do need to find the
inverse of a 3x3 matrix or matrix that's
bigger than that it is recommended to
use this calculator method to find it
calculating the inverse of a matrix
that's that big is very involved and
very meticulous now for this part of the
tutorial let's actually do an example
that applies inverse matrices to real
problems so let's take a look at example
3 so here we have a system of two linear
equations and if we want to solve them
rather than using substitution or
elimination method we could also use
matrices so what we need to do is take
this system of equations and change them
or convert them into a form that would
show this system as a matrix equation so
in very basic terms we want to have a
matrix with the coefficients of our
variables times a matrix that has our
variables in it and that's going to
equal a matrix with our constant values
so for example 3 if we were to set it up
like this we would first have our
coefficient matrix now in the first row
we want to have the coefficients that
are in equation 1 now in the first
column we want the X coefficient and in
column 2 we want the coefficient for our
Y and then similarly in the second row
we want to have the coefficients from
the second equation so we'll have the
coefficient for X in the second equation
in our first column here and then our
coefficient for Y also from the second
equation will be in the second column
here on the bottom row now when we set
up a matrix with our variables the
variables need to be in the same order
as they are with our coefficient matrix
but rather than having them in our
columns we're going to have them
separated out into different rows so for
example with this variable matrix the
first row will have our X and the second
row will have our Y now after
multiplying these two matrices together
we should get the same values as we do
in our system so for example in the
first row we have the equivalency or the
value that our first equation should
equal so since negative 2x minus y
should equal negative 4 we have negative
4 in our first row and then similarly as
well we want to have our constant value
from the second equation in our second
row so that's why 14 is there on the
bottom now so far what we have is our
set up which looks like this so this is
just another way to show the system of
equations but rather than having it as a
system we have it as a matrix equation
now if we want to solve this system or
solve this matrix equation we want to
find out what x and y are so essentially
what we need to do is isolate this
matrix that has our variables now in
order to do that we need to get rid of
this coefficient matrix here on the left
side now this is where inverse matrices
comes into play so if we want to cancel
out this coefficient matrix we need to
multiply it by its inverse now if we do
that on one side of an equation we also
have to do the same thing on the other
side of the equation as well
so our next step would be to multiply it
by the inverse of our coefficient matrix
and again we need to do that to both
sides to maintain the equality now like
I said earlier if we take the
coefficient matrix and multiply by its
inverse it will cancel out so now this
is gone from the left side so after
doing that we're left with this so at
this point we have isolated
matrix with our variables so in order to
actually find the values for x and y all
we need to do is take the inverse of
this matrix here and multiply it by our
constant matrix or the matrix with our
constant values now this inverse matrix
here probably looks familiar and it
should because it's actually the matrix
that we took the inverse of in example 1
of this tutorial so rather than going
through all that work again we could
just use the result that we found from
doing that so let's plug in that
resulting matrix and for our inverse so
now that we have everything set up let's
actually go ahead and solve for x and Y
now let's start with X first so if we
want to solve for X we need to look at
the first row and first column now for
our first matrix we'll do the first row
and then we'll multiply by the first
column so when we do that we get x
equals 1 times the negative 4 plus 0.5
times 14 and when we simplify this 1
times negative 4 gives us negative 4 and
then plus a 1/2 times 14 which gives us
a value of 7 so X's can equal negative 4
plus 7 which gives us a value of 3 so
when our resulting matrix in the first
row and first column will have a value
of 3 for our X variable now let's go on
to solve for y so now that we're looking
at Y we need to do this second row of
our first matrix times the first column
of our second matrix
so when we're solving for y we need do y
equals a negative 3 times negative 4
plus negative 1 times positive 14 now
when we simplify this a negative 3 times
a negative 4 gives us a positive 12 and
then plus a negative 1 times a positive
14 gives us a negative 14 so Y is going
to equal 12 plus a negative 14 which is
the same thing as 12 minus 14 so our Y
will have a value of negative 2 so in
the second row and first column of our
resulting matrix will have a value of
negative 2 so in the end when we solve
our variable matrix containing x and y
it turns out that the values end up
being this resulting matrix here so X is
going to be 3 and Y is going to be
negative 2 so that would be our solution
for this particular system of linear
equations so here we have a 2x2 matrix
of 6 5 negative 5 and negative 4 we want
to find its inverse well the way that we
would do this by creating an Augmented
matrix is we'll have our matrix on the
left side and on the right side we'll
have an identity matrix of the same size
so since our matrix was a 2x2 matrix the
identity matrix on the other side needs
to be 2x2
now at this point we would just want to
get the Augmented matrix into reduced
row echelon form so we'll need to do row
operations to accomplish that so the
first thing that we'll want to do is
change our first row so that we have a
value of 1 in the first row and first
column so we'll need to change that 6
now the way that we could do this is by
taking our second row and adding those
values to our fur
stro and when we do that we have six
plus the negative five in the first
column which gives us a 1 and then the
second column five plus a negative four
also gives us a 1 and then in the third
column we have a 1 plus 0 which is also
1 and in the last column we have 0 plus
1 which gives us a value of 1 and then
our second row will stay the same now
that we've done that we want all the
values below our 1 in the first column
to be 0 so we want to change this
negative 5 to be 0 now the way that we
could do that is by taking five times
the values in our first row and adding
them to the values in our second row and
when we do this the first row will stay
the same
but for our second row we'll do 5 times
1 which is 5 plus a negative 5 gives us
0 and then in the second column we have
5 times 1 which is 5 plus a negative 4
gives us a value of 1 and then in the
third column we have 5 times 1 which is
5 and then plus 0 gives us a value of 5
and then for our last column 5 times 1
is 5 plus 1 gives us a value of 6 now
that we have our augmented matrix in row
echelon form let's put it into reduced
row echelon form by making this one
become a zero so to do that we could
change our first row by taking the
opposite values of our entries in the
second row and then add them to the
entries in our first row and when we do
that in the first row in first column
will have a negative zero
which is just zero plus one will give us
a value of one then in the second column
negative one plus one will give us a
zero and then in the third column
negative five plus one gives us a
negative four and then the last column
negative six plus one gives us a
negative five and then our second row
will stay the same now we have our
matrix in reduced row echelon form with
this information we now know what the
inverse of our matrix will be all we
have to do is take the 2x2 matrix that's
on the right side of our vertical bars
in the Augmented matrix
so the inverse of the matrix
six five negative five and negative four
equals the result from our augmented
matrix negative four negative five five
and six now you know how to find the
inverse of matrices using an Augmented
matrix in its reduced row echelon form
and this method could be used for any
size matrix as long as it has an inverse