A matrix function such as and can be used to figure out expressions. By calculating and the values of X and Y can be solved for. . By following rules of multiplying matrices, this can be shown as =. Using we can conclude that = or. One can generalize a statement of a pattern that develops as the matrix goes on. The expression is as follows, =. The number 2 in the matrix comes from when the product of is solved for. The value of is twice the value of . The variable n represents what power the matrix is to, such as n = 2, 3, 4. We can now solve for the rest of the values of.
Further values of can be proven by the expression, can be proven with the same expression by slightly changed. Since and are negative, we much change the expression as to meet the demands of and .
Just to be sure the expression works again, we can find higher values of .
By using GDC, the values of and were double checked for accuracy.
By having the values of and , they can be used in the expression to further advance the knowledge of matrices. is the same as the expression Adding matrices would be just like adding + and +, by adding the values of the columns and rows used, the value of the product, that is trying to be solved, will be found. An example of finding what values that could be put in would be;
Any variable will do for , so let us make the value of to be simple.
And is the identity matrix of a 2 column by 2 rows, or a 2×2 matrix, which is.
Now knowing what the identity is, an expression can be formed. .
Let the value of , to further prove and show the expression
The expression of fits and is proven by the examples above.
Let and where and are constants. Let us use different values of and to calculate the values of .
Solving the first example, we can create an expression that should work. The expression for the value can be written as . The comes from multiplying it with , which is shown earlier on, . The comes from the constant of which we solve the value of and raise it to the n power.
Continuing using, will now be solved for, using the expression.
The same expression can be used for when , . The comes from multiplying it with as shown earlier comes from the constant of we solve the value of and raise it to the power. Since and need to be different constants, the value of in this example will equal .
Keeping in mind what and are, we can now find the expression for . By already having and solved for, the conclusion of , which is ). For example, by using GDC, the conclusion of the expression ) can be brought about by double checking answers. Using the knowledge learned from above, another example can be used.
Now, let us consider Since and are constants but of different values, that must be taken into consideration for .
Let and which is the same as which is the same as . Now since the step further would be to see what was , and using the GDC, the answers found can be double checked.
The general statement for the equations above is. The part comes from page 5 where the general equations of what and are. To test the validity of the general statement, different values of all the variables need to be used.
There is flaw to the general statements above. The matrices can only be multiplied exponentially by any whole integer number that is greater than 0. The matrices must meet specific demands. They must have matching dimensions to be multiplied. An example is . The inner parts of the matrices must match, which the example follows since it’s a 2×2 ï¿½ 2×2. The outer portions of the matrices are the resulting dimensions when multiplied. For instances, you cannot multiply 2×1 ï¿½ 2×3 since the inner numbers in bold do not match. The matrices used in all examples for finding general states have all been 2×2, which limits the general statements to only 2x2s. The general statements would not work for a 3×3 or any others besides a 2×2.
The general statement is . One would get to this general statement algebraically when multiplying or exponentially. The 2 in the equation is twice as much as the square numbers and that is where the number 2 comes from in the general statement. Since 2 receives less than the power and this is where the section of arrives from in the equation . When , and are given earlier on in the paper from their expressions that were found by solving various problems.