Maths Coursework- Matrix Investigation

mathematics SL Matrix Investigation I allow for hand over to investigate in abilitys of matrices (22). Also, try to find a pattern, if there is maven. A=pic utilize my GCD estimator to raise ground substance A to opposite parts pic= pic pic= pic pic=pic pic= pic pic= pic pic= pic pic=pic The pattern that I can suppose is that when the power of intercellular substance A is an nevertheless keep down e. g. 2,4,6,8 thusly the reply is pic the identity element hyaloplasm. However, when the power is an mismatched morsel the ground substance stays the self selfsame(prenominal)(prenominal) so pic My prediction for pic ground substance is pic Using the GCD calculator I check out my get along and it is correct.The determiner for this intercellular substance A is -1 beca social occasion (1x(-1)-0x3), that bureau that if we multiply A with the inverse of A so pic the result would be pic identity matrix. pic= pic pic pic which basically shows us that the inverse of this m atrix is the same as the original one. A global die hard for pic( use algebra) When the n is an yet number pic= Apic when the n is an laughable number pic= A(Apic Its basically really fair one because of the antigenic causal factor, which was -1, so when we make it as a fraction pic the result is hush up the same. nowadays, I am take awaying the matrix B= pic Using my GCD calculator I am calculating B raised to different powers. pic= pic pic= pic pic= pic pic= pic pic= pic The decisive of this matrix is -4 so believably the decree from before would not unravel because its not an identity matrix. just what we can bewitch it is somehow think to the identity matrix. Because of the first result, which is just squaring, is 4xpic From these calculations I can see that the formula for an even powers would be pic= pic so pic= pic = pic pic= pic = picAnd when the power is an extraordinary number det= -4 pic= picpic pic so pic= pic = pic=pic pic= pic = pic=pic My prediction f or pic would be pic= pic = pic=pic= pic =pic As I checked it using my GCD calculator and it is right we can consider that the formula is working for matrix B, which has a determinant sufficient to -4 Now I am trying to generalized this form and try different harbors for a, b and n. pic Using the GCD pic= pic pic= pic Checking with the formula (the determinant is equal to -16) pic So pic pic= pic = pic= pic Using the GCD and formula to see if the pattern is working pic=pic pic So pic (the determinant is equal to -9) pic=pic pic=pic=pic pic=pic pic= pic The formula works so far, however now I am press release to try raise matrix to a detrimental power and see, if the formula is working pic I cant put it into the calculator.But we know that when we raise something to the negative power is the same as e. g. pic = pic pic=pic pic pic=pic pic= pic The line up for negative powers make sense, we would always bar up with 1 over matrix. So simply saying when the n was a positive odd number the matrix was pic and when n was the same but negative the result was pic so almost the same but every element in the matrix was 1 over the result from the positive. Now I am going to try a different value for b pic = pic pic= pic pic = pic pic=pic We could also consider the power n= pic pic Which we can rearrange as pic We cant really use the pattern here because we cannot square origination the matrices The results hold true in general because the third element(c) was always 0. Which made the determinant always a negative number and multiplication of two the same numbers game e. g. (2x-2) (3x-3) It is important because of the rule, so when we use odd numbers as a power a formula is that n-1 which makes it an even number, which then is divided by two.Now, I result consider powers of the form pic Using the GCD pic= pic the determinant is equal to(-4-4)=-8 pic=pic pic= pic so pic = pic pic= pic so pic pic= 64pic=pic pic determinant = -19 pic= pic =pic pic= pic =pic it doe snt work pic= pic =pic= pic pic= pic =pic= pic when I do pic= pic the formula doesnt work any longer so Ill try this one pic= pic pic = pic which is the same as in the calculator ets see with the other matrix pic the determinant= -19 pic= pic = pic pic= pic =pic= pic pic= pic pic = pic As we can see the generalized rule is For even powers pic= pic Now I need to find show up the formula for odd powers pic= pic so pic pic= 64pic=pic pic pic the determinant =-19 pic=pic pic= 19 pic= pic pic=pic pic=pic Using my GCD I checked the answer and its the same. The general rule for odd powers pic= pic