Problem17:

Find an upper triangular matrix A, such that,

> ;

```                                 3   [1    30]
A  = [       ]
[0    -8]```

SOLUTION:

 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!

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>