Alg1131W12T2 - A quadratic matrix identity

Sunday, 27 May 2018

10:05 PM

The arithmetic of 2 x 2 matrices is actually an incredibly rich extension of ordinary arithmetic. Recall the product of two such matrices: ax + bz ay + bw cc + dz cy + dw In particular a and as an even more special case a 0 o d o a x 0 0 O w These last matrices are all multiples of the 2 x 2 identity matrix given by 1 Note: ax + bz ay + bw • the maple syntax for the matrix cc + dz cy + dw a*x+b*z I a*y+b*w > , < c*x+d*z I c*y+d*w >> • It is recommended that you include a space beftveen the less than symbol < and any letters to prevent your web browser from confusing it with an HTML tag

 

For the matrix -5 we compute that -5 A2 Hence we can write .42 in terms of A and I 1 o 1 where "1 — O and Sl — -35 This formula allows us to write .44 in terms of as A and I as: where r 2 147 O and s 2 a Note Maple syntax for the matrix c b d is<< a I b c

 

o The matrix B 1 has some interesting algebraic structure. Tick all of the following statements about B that are true. Here I refers to the 2 x 2 identity matrix. 1 B3 -1 -B -1

 

Consider the matricies A , and B • the smallest positive integer n such that An • the smallest positive integer n such that Bn Geometrically the transformation O a reflection in the x-axis O a reflection in the y-axis @ a reflection in the line y O a reflection in the line y O a rotation by 900 O a rotation by —900 Geometrically the transformation O a reflection in the x-axis O a reflection in the y-axis O a reflection in the line y O a reflection in the line y O a rotation by 900 @ a rotation by —900 We can also compute: Hence we are reminded that matrix multiplication is O -1 1 and I 1 o Then: 1 lis 2 lis 4 not commutative '

 

Recall that the Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, . . may be defined inductively via the rules FO = 0, Fl = 1 and Fn+l ¯ The linear relationship between these numbers can be represented by the matrix equation Furthermore In general Fn is the 1 1 top right Then O O 1 entry of the matrix 1 1 where r . A4 . A8 1 and s O .42 = rA + sl Hence we can easily compute higher powers, such as [enter your answer as a matrix]: FromA8 we see F8 is 21 2178309 1 Note: recall the Maple notation for 3 O . Similarly we can find F32 — 2 112 >,< 314 4

 

 

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