document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170702>Question 1170702</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let M=[{0,-2},{4,6}]. Find formulas for the entries of Mn, where n is a positive integer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851108 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851108\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's find the formula for the entries of $M^n$, where $M = \begin{bmatrix} 0 & -2 \\ 4 & 6 \end{bmatrix}$.\r<BR>\n" );
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document.write( "**1. Find the Eigenvalues and Eigenvectors of M**\r<BR>\n" );
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document.write( "* **Characteristic Equation:**<BR>\n" );
document.write( "   det(M - λI) = 0<BR>\n" );
document.write( "   det( $\begin{bmatrix} -\lambda & -2 \\ 4 & 6-\lambda \end{bmatrix}$ ) = 0<BR>\n" );
document.write( "   (-λ)(6 - λ) - (-2)(4) = 0<BR>\n" );
document.write( "   -6λ + λ² + 8 = 0<BR>\n" );
document.write( "   λ² - 6λ + 8 = 0<BR>\n" );
document.write( "   (λ - 4)(λ - 2) = 0\r<BR>\n" );
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document.write( "* **Eigenvalues:**<BR>\n" );
document.write( "   λ₁ = 4 and λ₂ = 2\r<BR>\n" );
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document.write( "* **Eigenvectors:**<BR>\n" );
document.write( "    * For λ₁ = 4:<BR>\n" );
document.write( "       (M - 4I)v₁ = 0<BR>\n" );
document.write( "       $\begin{bmatrix} -4 & -2 \\ 4 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$<BR>\n" );
document.write( "       -4x - 2y = 0<BR>\n" );
document.write( "       -2x - y = 0<BR>\n" );
document.write( "       y = -2x<BR>\n" );
document.write( "       Let x = 1, then y = -2.<BR>\n" );
document.write( "       v₁ = $\begin{bmatrix} 1 \\ -2 \end{bmatrix}$\r<BR>\n" );
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document.write( "    * For λ₂ = 2:<BR>\n" );
document.write( "       (M - 2I)v₂ = 0<BR>\n" );
document.write( "       $\begin{bmatrix} -2 & -2 \\ 4 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$<BR>\n" );
document.write( "       -2x - 2y = 0<BR>\n" );
document.write( "       x + y = 0<BR>\n" );
document.write( "       y = -x<BR>\n" );
document.write( "       Let x = 1, then y = -1.<BR>\n" );
document.write( "       v₂ = $\begin{bmatrix} 1 \\ -1 \end{bmatrix}$\r<BR>\n" );
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document.write( "**2. Form the Matrix P and Diagonal Matrix D**\r<BR>\n" );
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document.write( "* P = $\begin{bmatrix} 1 & 1 \\ -2 & -1 \end{bmatrix}$ (matrix of eigenvectors)<BR>\n" );
document.write( "* D = $\begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix}$ (diagonal matrix of eigenvalues)\r<BR>\n" );
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document.write( "**3. Find P⁻¹**\r<BR>\n" );
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document.write( "* det(P) = (1)(-1) - (1)(-2) = -1 + 2 = 1<BR>\n" );
document.write( "* P⁻¹ = (1/det(P)) $\begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$ = $\begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$\r<BR>\n" );
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document.write( "**4. Use the Formula Mⁿ = PDⁿP⁻¹**\r<BR>\n" );
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document.write( "* Dⁿ = $\begin{bmatrix} 4ⁿ & 0 \\ 0 & 2ⁿ \end{bmatrix}$\r<BR>\n" );
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document.write( "* Mⁿ = $\begin{bmatrix} 1 & 1 \\ -2 & -1 \end{bmatrix} \begin{bmatrix} 4ⁿ & 0 \\ 0 & 2ⁿ \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$\r<BR>\n" );
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document.write( "* Mⁿ = $\begin{bmatrix} 4ⁿ & 2ⁿ \\ -2(4ⁿ) & -2ⁿ \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$\r<BR>\n" );
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document.write( "* Mⁿ = $\begin{bmatrix} -4ⁿ + 2(2ⁿ) & -4ⁿ + 2ⁿ \\ 2(4ⁿ) - 2(2ⁿ) & 2(4ⁿ) - 2ⁿ \end{bmatrix}$\r<BR>\n" );
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document.write( "* Mⁿ = $\begin{bmatrix} -4ⁿ + 2^(n+1) & -4ⁿ + 2ⁿ \\ 2(4ⁿ) - 2^(n+1) & 2(4ⁿ) - 2ⁿ \end{bmatrix}$\r<BR>\n" );
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document.write( "**Formulas for the Entries:**\r<BR>\n" );
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document.write( "* Mⁿ₁₁ = -4ⁿ + 2^(n+1)<BR>\n" );
document.write( "* Mⁿ₁₂ = -4ⁿ + 2ⁿ<BR>\n" );
document.write( "* Mⁿ₂₁ = 2(4ⁿ) - 2^(n+1)<BR>\n" );
document.write( "* Mⁿ₂₂ = 2(4ⁿ) - 2ⁿ<BR>\n" );
document.write( " </SPAN>\n" );
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