Welcome to block 2 Linear Algebra.
Now we will practice with matrices.
First of all, an example,
with addition, subtraction and
scalar multiplication of matrices as the most elementary example.
Let be A, B, C, and D
these matrices.
Now calculate A+B, C-D,
C+A and 5A. Go on!
I'm going to give the solution
but you can work, stop de video
and then check the solution.
Then A+B
is the sum of two matrices.
See that for example,
the solution c11
is the sum of
of two and one.
That's three.
In the second column,
minus three plus
zero is minus three and so on
with all the numbers.
The second example,
C-D is the same,
the first will be seven minus six is one,
the second column, (-3)-0 = -3
and so on.
Remember the dimensions of matrix C
we demand to have C+A,
we can't do that because C and A
doesn't have the same dimension.
So it's not possible to do the sum
what we demand in c)
and d) calculate 5A
The first column, five by two is 10
5*0=0 and 5*7=35,
so these are the basic operations
with matrices.
Now matrix multiplication,
let be A and B these two matrices
A a 2x2 matrix and
B a 2x3 matrix
we can do the multiplication
because the number of rows of B,
is the same number of the columns of A.
Let's calculate matrix C.
You can see the first
number in red is c11
is the inner product of the first row of A,
and the first column of B. See the numbers in different colors
you need to
multiply 1 by -1
and sum with three by zero.
And the result is the number c11 value -1.
And similarly we calculate the other
entries.
You can see the row and column is changing
and it depends
on the row
and column corresponding number.
So you can calculate
and then check
the solution.
Let's try,
you can calculate, stop the video
and then check the solution.
So we have the product of these two matrices.
Let's now calculate
the inverse of a matrix n by n,
in this case three by three.
Let's find the inverse of this matrix
and proof that this matrix is really the inverse,
so we need to proof this property
that the product of the matrix A and the inverse
in both directions of multiplication
is the identity matrix.
Just as we said in the previous video,
we began with the three by six matrix.
In the left we have the matrix A,
and in the right the identity matrix.
We transform this matrix
just doing
row operations.
See that
if we multiply
the first row by two
and we do the first operation
then we have
the next row
with these values.
And we continue doing different row operations
and now let's see, we have in the left the identity
and in the right we have the inverse matrix.
You have the inverse of the given matrix.
And to prove that
this is the matrix, the inverse,
to prove that we will
do the product of matrices.
This is another example of matrices multiplication.
Remember the colors red and green.
So do the operations
and you can see that, in both sense of multiplication,
you have the identity matrix.
So this is really the inverse of the
given matrix.
You need to calculate it slowly and
stop de video and go on.
Okay, here I will show you
how to calculate quickly an inverse
of a matrix, if you need.
There are a lot of web products nowadays that calculate inverse matrices.
Here you have one that you have in the references.
Is CalcMe of wiris
you put your entries in the matrix,
and calculate the inverse matrix. You can calculate with numbers
or with letters, if you need.
So matrix operations is important
also to solve linear systems of equations.
Let's solve this system,
linear system of equations.
First of all, we need to have the matrix
that represent the system
You put the entries, the coefficients
and the independent term
as the last column of the matrix.
With this new matrix
we do row operations
of as we saw before, and
we have this new matrix more simply.
This matrix is associated with this new system.
See that for example, with the last equation we know that
z=1 and then
we can have y, and with the first equation
we have x. So
with these matrix operations
we can solve the system and the solution of this particular system
is x=3, y=2 and z=1.
Here you have the references
for more information, theory,
examples on how to calculate. I think is important,
nowadays we have lot of products
but here you have some of the most important.
However, there are many products to calculate
with matrices computationally.
Enjoy!