Question

if matrix A=[ 0 0 0 1] then the matrix give n by B=I+A+A^2+... +A^k is​

Answer 1

Answer:

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

$A = \displaystyle \: \begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix}$

Now

${A}^{2}$

$= {A}\times A$

$= \displaystyle \: \begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix}$

$= \begin{pmatrix} 0 + 0 & 0 + 0\\ 0 + 0 & 1 + 0 \end{pmatrix}$

$= \displaystyle\begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix}$

So

${A}^{2} = {A}$

So

${A}^{3} = {A}^{2} \times {A}^{} ={A}\times {A}^{} = {A}^{2} = {A}$

Similarly

${A}^{k} = {A}$

HENCE

$B = I + A + {A}^{2} + {A}^{3} + ...... + {A}^{k}$

$= I + A + {A}+ {A} + ...... + {A}$

$= I + k \: A$

$= \displaystyle\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} + k \displaystyle\begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix}$

$= \displaystyle\begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} + \displaystyle\begin{pmatrix} 0 & 0\\ 0 & k \end{pmatrix}$

$= \displaystyle\begin{pmatrix} 1 & 0\\ 0 & k + 1 \end{pmatrix}$