The,;, shown below were calculated by using a GDC.
By analyzing the resulting matrices of different powers of X and Y we can develop an expression for and which are shown below.
Now to generate an expression for we must first develop the matrix resulting from X+Y so we can then use different powers for (X+Y) and hence generate the expression for
X + Y= + =
(X + Y)=
By analyzing the resulting matrices of different powers of (X+Y) we can develop an expression for which is shown below.
3)
Let A = aX and B = bY, where a and b are constants.
Use different values of a and b to calculate, ; , .
The values that we are going to use for a are 3 and 8, and the ones for b are 2 and 5.
(i)Matrix A when a is 3 would be equal to 3X.
Using a GDC we can now calculate, ; having in count that a is 3 in this case.
(ii)Matrix A when a is 8 would be equal to 8X.
Using a GDC we can now calculate,; having in count that a is 8 in this case.
(iii)Matrix B when b is 2 would be equal to 2Y.
Using a GDC we can now calculate,; having in count that b is 2 in this case.
(iv)Matrix B when b is 5 would be equal to 2Y.
Using a GDC we can now calculate,; having in count that b is 5 in this case.
By considering different integer powers of A and B find an expression for, and.
Since we have already calculated the resulting matrices using 2, 3 and 4 as integer power values of A and B in the first part of this question we can use the previously calculated matrices to find and expression for and .
We must then remember that both matrices have the same origin since A by definition is aX, so is the same as which can also be expressed as), we can check this by diving each element of both of the calculated previously by their respective values of and the result should be equal to.
In the case of a being 8 ,
While in the case of a being 3 ,
So if can be expressed like then can be expressed like which can be expressed in a matrix form, like it is shown below.
=
=
To find an expression for we can do the same process done to generate the expression for , so we can star by stating that is by definition, so should be equal to which can also be expressed as), we can check this by diving each element of both of the calculated previously by their respective values of and the result should be equal to.
In the case of b being 2 ,
While in the case of b being 5 ,
So if can be expressed like then can be expressed like which can be expressed in a matrix form, like it is shown below.
=
=
The expression for can be found in the same way that we found the expressions for and , we know that by definition A is aX and is so is the same as. Starting from this we can develop an expression for as shown below.
=
The expression for is then
4)
Now consider
Show that
We know that matrix = so know we just need to add matrix A to matrix B and see if the resulting matrix is the same as , taking in count that by definition A is aX and is so and since in this case the value of is 1 we can use the expression developed in question 3 and we can see that certainly . However this is also illustrated below.
To show that we first have to calculate and.
andcan be calculated by using the previously developed expression for and
==
==
=
Hence find the general statement that expresses in terms of and .
From the previous question we can deduce that so in order to find a general statement for we could just add up the expressions developed for and and that should give us the general statement.
=
Since both and are being multiplied by we can take it out of bothandand leave it as a scalar multiplying the adition of the matrices as shown below.
=
=
=
And so the general statement for would be:
=
We can prove this by seeing that if when we replace a and b with 3 and 2 respectively it should give us the add up of A + B when n has a value of 1.
We have then
=
=
Now if we use our general statement it should give use the same when we replace the values of a, b and n with the ones used to develop
=
For this case it worked, but to check that it is a general rule I am going to do it wi the following value for a, b and n.
a=8
b=2
n=3
Since we have already calculated and having the previously established values we can see if the addition of andin this case is also equal to the general statement.
We have then
=
=
Now if we use our general statement it should give use the same when we replace the values of a, b and n with the ones used to develop
==
==
Once again the statement worked and therefore I can consider it a general statement.
CONCLUSION
Nb b
- The general statement developed did worked correctly and was checked two times with different values for a, b and n.
-
The relationship between the matrix with the matrices and was found and it was expressed in matrix form in question 4.
-
We were able to generate expressions for specific matrices powered by an unknown n integer and having developed those expressions we were able to generate a general statement.
EVALUATION
In my opinion the project was developed in a good and coherent manner, there weren’t any serious problems or difficulties during the processes and the results obtained were satisfactory, since I was able to generate the general statement and the other expressions for the different matrices powered by an n integer.
During the process I used all my knowledge about matrices which helped me to develop a good understanding of the questions in order to obtain the different answer.
However I think that the information in some parts is not presented in an organized way, this could have been improved with the use of tables to record the information in a more suitable place.
