Question

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[tex]If \: A \: = \begin{array}{| c c c |}
\ \textless \ br /\ \textgreater \ 0 & 1 & 0 \\
\ \textless \ br /\ \textgreater \ 0 & 0 & 1 \\
\ \textless \ br /\ \textgreater \ 0 & 0 & 0
\ \textless \ br /\ \textgreater \ \end{array}\: and \: I \: is \: the \: limit \: matrix \: of \: order \: 3 \: then \: prove \: that \: {A}^{3} = PI + qA + r {A }^{2} ....[/tex]

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Answer 1

Answer :-

Some error correct question :-

$A = \left[\begin{array}{ccc}0&1&0\\0&0&1\\p&q&r\end{array}\right]$

And I is the unit matrix of order 3

Then we have to show

$A^3 = pI + qA + rA^2$

Now

$A^2 = \left[\begin{array}{ccc}0&1&0\\0&0&1\\p&q&r\end{array}\right] \times \left[\begin{array}{ccc}0&1&0\\0&0&1\\p&q&r\end{array}\right]\\\\ = \left[\begin{array}{ccc}0+ 0 + 0 & 0 + 0 + 0 & 0 + 1 + 0 \\0+0+p&0+0+q&0+0+r\\0+0+rp& p +0+rq & 0+q+r^2\end{array}\right]\\\\= \left[\begin{array}{ccc}0&0&1\\p&q&r\\rp&p+rq&q+r^2\end{array}\right]$

Now

$A^3 = \left[\begin{array}{ccc}0&0&1\\p&q&r\\rp&p+rq&q+r^2\end{array}\right] \times \left[\begin{array}{ccc}0&1&0\\0&0&1\\p&q&r\end{array}\right]\\\\=\left[\begin{array}{ccc}0+0+p&0+0+q&0+0+r\\0+0+rp&p+0+rq&0+q+r^2\\0+0+pq+r^2p&rp+0+q^2+r^2q& 0+p+rq+rq+r^3\end{array}\right] \\\\=\left[\begin{array}{ccc}p&q&r\\rp&p+rq&q+r^2\\pq+r^2p&rp+q^2+r^2q&p+2rq+r^3\end{array}\right] ...(1)$

Now :-

= pI + qA + rA²

$= p \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] +q\left[\begin{array}{ccc}0&1&0\\0&0&1\\p&q&r\end{array}\right] + r\left[\begin{array}{ccc}0&0&1\\p&q&r\\rp&p+rq&q+r^2\end{array}\right]\\\\= \left[\begin{array}{ccc}p&0&0\\0&p&0\\0&0&p\end{array}\right] +\left[\begin{array}{ccc}0&q&0\\0&0&q\\pq&q^2&rq\end{array}\right] + \left[\begin{array}{ccc}0&0&r\\pr&qr&r^2\\r^2p&pr+r^2q&qr+ r^3\end{array}\right]$

$= \left[\begin{array}{ccc}p&q&r\\rp&p+rq&q+r^2\\pq+r^2p&rp+q^2+r^2q&p+2rq+r^3\end{array}\right] ...(2)$

Now from (1) and (2)

LHS = RHS

Hence Shown

Answer 2

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