Problem17:Find an upper triangular matrix A, such that, 
> ;
3 [1 30] A = [ ] [0 8]

Just multiply a general 2 by 2 upper triangular matrix to see the pattern. 
> A := matrix(2,2,[a,b,0,c]);
[a b] A := [ ] [0 c]> A2 := evalm(A^2);
[ 2 ] [a a b + b c] A2 := [ ] [ 2 ] [0 c ]> A3 := evalm(A^3);
[ 3 2 ] [a a b + (a b + b c) c] A3 := [ ] [ 3 ] [0 c ]
Hence, we need to find a,b and c so that A3 coincides with the matrix A^3 given in the problem. The three equations are, 
> Eq1 := a^3 = 1; Eq2 := c^3 = 8; Eq3 := b*(a^2+a*c+c^2) = 30;
3 Eq1 := a = 1 3 Eq2 := c = 8 2 2 Eq3 := b (a + c a + c ) = 30
and the solution is straight forward, 
> a := 1: c := 2: Eq3;
3 b = 30
and b = 10. Let's check, 
> matrix(2,2,[1,10,0,2])^3;
[1 10]3 [ ] [0 2]
evaluates to: 
> evalm(%);
[1 30] [ ] [0 8]
cool! 