document.write( "Question 877488: Hello, my name is Alec. This homework problem has been stumping me.\r
\n" ); document.write( "\n" ); document.write( "I am given this matrix:\r
\n" ); document.write( "\n" ); document.write( "Matrix M = {{1,0,2},{-1,1,2},{-1,-1,2}}\r
\n" ); document.write( "\n" ); document.write( "Then asked to solve this equation:\r
\n" ); document.write( "\n" ); document.write( "M^3 + xM^2 + yM + zI = 0\r
\n" ); document.write( "\n" ); document.write( "I is the 3x3 identity matrix.
\n" ); document.write( "Any hints on how to get started would be appreciated! Thank you \n" ); document.write( "
Algebra.Com's Answer #529362 by rothauserc(4718)\"\" \"About 
You can put this solution on YOUR website!
student says that x, y, z are constants
\n" ); document.write( "Calculate M^2 and M^3, matrix multiplication is row times column
\n" ); document.write( "M^2 = | 1 0 2 | * | 1 0 2 |
\n" ); document.write( " | -1 1 2 | | -1 1 2 |
\n" ); document.write( " | -1 -1 2 | | -1 -1 2 |
\n" ); document.write( "M^2 = | 1+0-2 0+0-2 2+0+4 |
\n" ); document.write( " | -1-1-2 0+1-2 -2+2+4 |
\n" ); document.write( " | -1+1-2 0-1-2 -2-2+4 |
\n" ); document.write( "M^2 = | -1 -2 6 |
\n" ); document.write( " | -4 -1 4 |
\n" ); document.write( " | -2 -3 0 |
\n" ); document.write( "M^3 = M^2 * M
\n" ); document.write( "M^3 = | -1 -2 6 | * | 1 0 2 |
\n" ); document.write( " | -4 -1 4 | | -1 1 2 |
\n" ); document.write( " | -2 -3 0 | | -1 -1 2 |
\n" ); document.write( "M^3 = | -1+2-6 0-2-6 -2-4+12 |
\n" ); document.write( " | -4+1-4 0-1-4 -8-2+8 |
\n" ); document.write( " | -2+3+0 0-3+0 -4-6+0 |
\n" ); document.write( "m^3 = | -5 -8 6 |
\n" ); document.write( " | -7 -5 -2 |
\n" ); document.write( " | 1 -3 -10 |
\n" ); document.write( "now calculate xM^2, yM and zI
\n" ); document.write( "xM^2 = x * | -1 -2 6 |
\n" ); document.write( " | -4 -1 4 |
\n" ); document.write( " | -2 -3 0 |
\n" ); document.write( "xM^2 = | -x -2x 6x |
\n" ); document.write( " | -4x -x 4x |
\n" ); document.write( " | -2x -3x 0 |
\n" ); document.write( "yM = y * | 1 0 2 |
\n" ); document.write( " | -1 1 2 |
\n" ); document.write( " | -1 -1 2 |
\n" ); document.write( "yM = | y 0 2y |
\n" ); document.write( " | -y y 2y |
\n" ); document.write( " | -y -y 2y |
\n" ); document.write( "zI = z * | 1 0 0 |
\n" ); document.write( " | 0 1 0 |
\n" ); document.write( " | 0 0 1 |
\n" ); document.write( "zI = | z 0 0 |
\n" ); document.write( " | 0 z 0 |
\n" ); document.write( " | 0 0 z |
\n" ); document.write( "we know
\n" ); document.write( "M^3 + xM^2 + yM + zI = 0
\n" ); document.write( "| -5 -8 6 | + | -x -2x 6x | + | y 0 2y | + | z 0 0 | = 0
\n" ); document.write( "| -7 -5 -2 | | -4x -x 4x | | -y y 2y | | 0 z 0 |
\n" ); document.write( "| 1 -3 -10 | | -2x -3x 0 | | -y -y 2y | | 0 0 z |
\n" ); document.write( "we now add the elements in our matrices
\n" ); document.write( "| -5-x+y+z -8-2x+0+0 6+6x+2y+0 | = 0
\n" ); document.write( "| -7-4x-y+0 -5-x+y+z -2+4x+2y+0 |
\n" ); document.write( "| 1+2x-y+0 -3-3x-y+0 -10+0+2y+z |
\n" ); document.write( "rewrite eliminating the 0's
\n" ); document.write( "| -5-x+y+z -8-2x 6+6x+2y | = 0
\n" ); document.write( "| -7-4x-y -5-x+y+z -2+4x+2y |
\n" ); document.write( "| 1-2x-y -3-3x-y -10+2y+z |
\n" ); document.write( "we see that y = 9, x = -4, z = -8\r
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