Not more than Infinity

Algebra Level 4

If A = [ 1 6 1 3 − 5 6 4 3 ] A=\begin{bmatrix} \frac{1}{6} & \frac{1}{3} \\ -\frac{5}{6} & \frac{4}{3} \end{bmatrix} and B = A ∞ B=A^\infty , then the Trace of B , T r ( B ) = ? Tr(B)=?

1 4 5 6

Anik Saha by Anik Saha , 19, Bangladesh

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1 solution

Mark Hennings
Dec 6, 2020

The matrix A A has eigenvalues 1 , 1 2 1,\tfrac12 , so there exists a nonsingular matrix P P such that P A P − 1 = ( 1 0 0 1 2 ) P A n P − 1 = ( 1 0 0 2 − n ) PAP^{-1} \; = \; \left(\begin{array}{cc} 1 & 0 \\ 0 & \tfrac12\end{array}\right) \hspace{2cm} PA^nP^{-1} \; = \; \left(\begin{array}{cc} 1 & 0 \\ 0 & 2^{-n}\end{array}\right) and hence B = lim ⁡ n → ∞ A n = P − 1 ( 1 0 0 0 ) P B \; = \; \lim_{n \to \infty} A^n \; = \; P^{-1}\left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)P so that T r [ B ] = T r [ ( 1 0 0 0 ) ] = 1 \mathrm{Tr}[B] \; = \; \mathrm{Tr}\left[\left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)\right] \; = \; \boxed{1}

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