Answer to Question #223067 in Linear Algebra for Haroun

Question #223067

A is a 2x2 matrix

Let A = {(2) (0) (0) (-1)}. Find A32

1
Expert's answer
2021-08-04T18:11:44-0400
A=(2001)A=\begin{pmatrix} 2 & 0 \\ 0 & -1 \end{pmatrix}

A2=(2001)(2001)=(2200(1)2)A^2=\begin{pmatrix} 2 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 2 & 0 \\ 0 & -1 \end{pmatrix}=\begin{pmatrix} 2^2 & 0 \\ 0 & (-1)^2 \end{pmatrix}

=(4001)=\begin{pmatrix} 4 & 0 \\ 0 & 1 \end{pmatrix}

A4=(A2)2=(4001)(4001)=(2400(1)4)A^4=(A^2)^2=\begin{pmatrix} 4 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 4 & 0 \\ 0 & 1 \end{pmatrix}=\begin{pmatrix} 2^4 & 0 \\ 0 & (-1)^4 \end{pmatrix}

=(16001)=\begin{pmatrix} 16 & 0 \\ 0 & 1 \end{pmatrix}

A8=(A4)2=(16001)(16001)=(2800(1)8)A^8=(A^4)^2=\begin{pmatrix} 16 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 16 & 0 \\ 0 & 1 \end{pmatrix}=\begin{pmatrix} 2^8 & 0 \\ 0 & (-1)^8 \end{pmatrix}

=(256001)=\begin{pmatrix} 256 & 0 \\ 0 & 1 \end{pmatrix}


A16=(A8)2=(256001)(256001)=(21600(1)16)A^{16}=(A^8)^2=\begin{pmatrix} 256 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 256 & 0 \\ 0 & 1 \end{pmatrix}=\begin{pmatrix} 2^{16} & 0 \\ 0 & (-1)^{16} \end{pmatrix}

=(65536001)=\begin{pmatrix} 65536 & 0 \\ 0 & 1 \end{pmatrix}


A32=(A16)2=(65536001)(65536001)A^{32}=(A^{16})^2=\begin{pmatrix} 65536 & 0 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 65536 & 0 \\ 0 & 1 \end{pmatrix}

=(23200(1)32)=(4294967296001)=\begin{pmatrix} 2^{32} & 0 \\ 0 & (-1)^{32} \end{pmatrix}=\begin{pmatrix} 4294967296 & 0 \\ 0 & 1 \end{pmatrix}



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