Trace Of Matrix Power

Algebra Level 2

Consider the $2$ -by- $2$ matrix $A = \begin{pmatrix} 5 & 1 \\ -1 & 3 \end{pmatrix}.$ What is the trace of $A^{100}?$

Note: The trace of a $2$ -by- $2$ matrix is defined as $\text{Tr}\begin{pmatrix} a & b \\ c& d \end{pmatrix} = a+d\; .$

2^{100}

2^{101}

2^{200}

2^{201}

3 solutions

Otto Bretscher
Apr 15, 2016

The eigenvalues of $A$ are 4, 4, so that the eigenvalues of $A^{100}$ are $4^{100},4^{100}$ and the trace of $A^{100}$ is $2\times 4^{100}=\boxed{2^{201}}$

Victor Moreno Diaz
Jul 29, 2016

A little few calculations give us $\text{tr}(A)=2^3$ , $\text{tr}(A^2)=2^5$ , $\text{tr}(A^3)=2^7$ ... So, if we define the trace sequence $a_n:=2n+1$ the question is to find $a_{100}$ .

Gabriele Manganelli
Dec 28, 2017

A is similar to the matrix J which has a11=4 , a12=1, a21=0, a22=4. Letting P and P^-1 be the matrices of change of basis, A=PJP^-1. A^100=(PJP^-1)(PJP^-1)...=PJ^100P^-1 since PP^-1=Identity Matrix.

Diagonalization

Ganishk Dhanakodi - 1 year ago

Trace Of Matrix Power

3 solutions

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