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<head><title>Matrix ^ ZZ -- power</title>
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<div><h1>Matrix ^ ZZ -- power</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>f^n</tt></div>
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<li><span>Operator: <a href="_^.html" title="a binary operator, usually used for powers">^</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span></span></li>
<li><span><tt>n</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, <tt>f^n</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : R = ZZ/7[x]/(x^6-3*x-4)
o1 = R
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : f = matrix{{x,x+1},{x-1,2*x}}
o2 = | x x+1 |
| x-1 2x |
2 2
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : f^2
o3 = | 2x2-1 3x2+3x |
| 3x2-3x -2x2-1 |
2 2
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : f^1000
o4 = | 3x5-2x4-2x2+2x+3 -2x5+x3-x2+x+1 |
| x5+x4-2x2-3x+1 -x5+2x4-3x3+x-3 |
2 2
o4 : Matrix R <--- R</pre>
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If the matrix is invertible, then f^-1 is the inverse.<table class="examples"><tr><td><pre>i5 : M = matrix(QQ,{{1,2,3},{1,5,9},{8,3,1}})
o5 = | 1 2 3 |
| 1 5 9 |
| 8 3 1 |
3 3
o5 : Matrix QQ <--- QQ</pre>
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<tr><td><pre>i6 : det M
o6 = 9
o6 : QQ</pre>
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<tr><td><pre>i7 : M^-1
o7 = | -22/9 7/9 1/3 |
| 71/9 -23/9 -2/3 |
| -37/9 13/9 1/3 |
3 3
o7 : Matrix QQ <--- QQ</pre>
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<tr><td><pre>i8 : M^-1 * M
o8 = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |
3 3
o8 : Matrix QQ <--- QQ</pre>
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<tr><td><pre>i9 : R = QQ[x]
o9 = R
o9 : PolynomialRing</pre>
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<tr><td><pre>i10 : N = matrix{{x^3,x+1},{x^2-x+1,1}}
o10 = | x3 x+1 |
| x2-x+1 1 |
2 2
o10 : Matrix R <--- R</pre>
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<tr><td><pre>i11 : det N
o11 = -1
o11 : R</pre>
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<tr><td><pre>i12 : N^-1
o12 = {3} | -1 x+1 |
{1} | x2-x+1 -x3 |
2 2
o12 : Matrix R <--- R</pre>
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<tr><td><pre>i13 : N^-1 * N
o13 = {3} | 1 0 |
{1} | 0 1 |
2 2
o13 : Matrix R <--- R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_determinant.html" title="determinant of a matrix">determinant</a> -- determinant of a matrix</span></li>
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