Matrix question?

Nappy

Hey Im studying for a linear Maths exam and am stuck on a question, Im studying upper and lower triangular matrices and I am asked to find the matrix that satisfies:

A3 = | 1 30 |
| 0 -8 | , where 3= cubed, Ive been multiplying

|a b | by itself 3 times and tring to solve the equations for a, b , c and d
|c d |

but getting no where??

Nappy

sorry got it as c would = 0 if upper triangular,

thanks anyway!

tolosenc

Assuming A is:
a b
c d

Then:

(a^3) + 2.a.b.c + b.c.d = 1
(a^2).b + (b^2).c + (d^2).b + a.b.d = 30
(a^2).c + a.c.d + c.(d^2) + (c^2)b = 0
a.b.c + 2.b.c.d + (d^3) = -8

Solve simultaneously for each. The 3rd one will make it kinda handy, I'd imagine.

Michael Collins

It's even simpler than that actually. Take a 2 x 2 triangular matrix M

M=[a  b]
  [0  d]

If we raise this to a power of n we get

M^n = [a^n    x ]
      [0     d^n]

where x is some more complicated expression involving a, b and d.

The extention of this to n x n triangular matrices is also true.

So in this case we can immediately say that
a^3 = 1 => a = 1
d^3 = -8 => d = -2

So we have

M = [1  x]
    [0  2]

and it remains to find x. So just multiply to get M^3 (this is extra easy as it's triangular):

M^3 = [1  3x]
      [0  -8]

Now by looking at M^3 given in the question it's obvious that x = 10. So finally we have that

M = [1  10]
    [0  -2]