SOLUTION: 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

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: 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       Log On


   

Question 877521: 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. x, y, and z are constants
Any hints on how to get started would be appreciated! Thank you
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
M=%28matrix%283%2C3%2C1%2C0%2C2%2C-1%2C1%2C2%2C-1%2C-1%2C2%29%29
yM=+%28matrix%283%2C3%2Cy%2C0%2C2y%2C-y%2Cy%2C2y%2C-y%2C-y%2C2y%29%29+
M%5E2=%28matrix%283%2C3%2C-1%2C-2%2C6%2C-4%2C-1%2C4%2C-2%2C-3%2C0%29%29
xM%5E2=+%28matrix%283%2C3%2C-x%2C-2x%2C6x%2C-4x%2C-x%2C4x%2C-2x%2C-3x%2C0%29%29+
M%5E3=%28matrix%283%2C3%2C-5%2C-8%2C6%2C-7%2C-5%2C-2%2C1%2C-3%2C-10%29%29
I=%28matrix%283%2C3%2C1%2C0%2C0%2C0%2C1%2C0%2C0%2C0%2C1%29%29
zI=+%28matrix%283%2C3%2Cz%2C0%2C0%2C0%2Cz%2C0%2C0%2C0%2Cz%29%29+
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M%5E3%2BxM%5E2%2ByM%2BzI=0


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Using the first row of matrix elements,
-x%2By%2Bz=5
-2x=8
6x%2B2y=-6
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Which leads to
x=-4
y=9
z=-8
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All three rows yield the same results so it is consistent.