Maths Coursework- Matrix Investigation

Maths SL Matrix Investigation I will try to investigate in powers of matrices (22). Also, try to find a pattern, if there is one. A=pic Using my GCD calculator to raise ground substance A to different powers pic= pic pic= pic pic=pic pic= pic pic= pic pic= pic pic=pic The pattern that I lav sympathise is that when the power of matrix A is an even jump e. g. 2,4,6,8 then the result is pic the identity matrix. However, when the power is an odd trope the matrix stays the same so pic My prediction for pic matrix is pic Using the GCD calculator I checked my answer and it is correct.The determinant for this matrix A is -1 beca hold (1x(-1)-0x3), that means that if we multiply A with the inverse of A so pic the result would be pic identity matrix. pic= pic pic pic which fundamentally shows us that the inverse of this matrix is the same as the original one. A oecumenical practice for pic( exploitation algebra) When the n is an even recite pic= Apic when the n is an odd number pic= A (Apic Its basically really simple one because of the determinant, which was -1, so when we make it as a fraction pic the result is still the same.Now, I am considering 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 determinant of this matrix is -4 so probably the formula from in the lead would not work because its not an identity matrix. But what we can see it is somehow related 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 odd number det= -4 pic= picpic pic so pic= pic = pic=pic pic= pic = pic=pic My prediction for 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 equal to -4 Now I am trying to generalised this rule and try different values 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 going to try raise matrix to a negative 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 rule for negative powers make sense, we would always end 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 root 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 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 will 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 doesnt work pic= pic =pic= pic pic= pic =pic= pic when I do pic= pic the formula doesnt work anymore 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 p ic= 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 out 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