Question 877488: Hello, my name is Alec. This homework problem has been stumping me.
I am given this matrix:
Matrix M = {{1,0,2},{-1,1,2},{-1,-1,2}}
Then asked to solve this equation:
M^3 + xM^2 + yM + zI = 0
I is the 3x3 identity matrix.
Any hints on how to get started would be appreciated! Thank you
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! 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
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