Question 877488
student says that x, y, z are constants
Calculate M^2 and M^3, matrix multiplication is row times column
M^2 = |  1  0  2 |   *   |  1  0  2 | 
      | -1  1  2 |       | -1  1  2 |
      | -1 -1  2 |       | -1 -1  2 |
M^2 = |  1+0-2 0+0-2  2+0+4 |
      | -1-1-2 0+1-2 -2+2+4 |
      | -1+1-2 0-1-2 -2-2+4 |
M^2 = |  -1  -2  6  |
      |  -4  -1  4  |
      |  -2  -3  0  |
M^3 = M^2 * M
M^3 = |  -1  -2  6  |   *  |  1  0  2 |  
      |  -4  -1  4  |      | -1  1  2 |
      |  -2  -3  0  |      | -1 -1  2 | 
M^3 = | -1+2-6 0-2-6 -2-4+12  |
      | -4+1-4 0-1-4 -8-2+8   |
      | -2+3+0 0-3+0 -4-6+0   |
m^3 = | -5  -8  6  |
      | -7  -5 -2  |
      |  1  -3 -10 |
now calculate xM^2, yM and zI
xM^2 = x *  |  -1  -2  6  |
            |  -4  -1  4  |
            |  -2  -3  0  |
xM^2 =  | -x   -2x  6x  |
        | -4x  -x   4x  |
        | -2x  -3x  0   |
yM   =  y  *  |  1  0  2 |
              | -1  1  2 |
              | -1 -1  2 |
yM = |  y  0 2y  |
     | -y  y 2y  |
     | -y -y 2y  |
zI = z  *  | 1 0 0 |
           | 0 1 0 |
           | 0 0 1 |
zI = | z 0 0 |
     | 0 z 0 |
     | 0 0 z |
we know
M^3 + xM^2 + yM + zI = 0 
| -5  -8  6  |  +  | -x  -2x  6x  |  +  |  y  0 2y  |  + | z 0 0 |   =  0
| -7  -5 -2  |     | -4x  -x  4x  |     | -y  y 2y  |    | 0 z 0 |
|  1  -3 -10 |     | -2x   -3x  0 |     | -y -y 2y  |    | 0 0 z |
we now add the elements in our matrices
| -5-x+y+z  -8-2x+0+0  6+6x+2y+0  |   =  0
| -7-4x-y+0  -5-x+y+z -2+4x+2y+0  |
| 1+2x-y+0   -3-3x-y+0 -10+0+2y+z |
rewrite eliminating the 0's
| -5-x+y+z  -8-2x      6+6x+2y   |  = 0
| -7-4x-y   -5-x+y+z  -2+4x+2y   |
|  1-2x-y   -3-3x-y  -10+2y+z    |
we see that y = 9, x = -4, z = -8