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If $A = \left[ {\begin{array}{*{20}{c}}1&2\\0&1\end{array}} \right],$then ${A^n} = $
$\left[ {\begin{array}{*{20}{c}}1&{2n}\\0&1\end{array}} \right]$
$\left[ {\begin{array}{*{20}{c}}2&n\\0&1\end{array}} \right]$
$\left[ {\begin{array}{*{20}{c}}1&{2n}\\0&{ - 1}\end{array}} \right]$
$\left[ {\begin{array}{*{20}{c}}1&{2n}\\1&0\end{array}} \right]$
Solution
(a) ${A^2} = \left[ {\begin{array}{*{20}{c}}1&2\\0&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}1&2\\0&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&4\\0&1\end{array}} \right]$ and ${A^3} = {A^2}A$.
= $\left[ {\,\begin{array}{*{20}{c}}1&4\\0&1\end{array}\,} \right]\,\,\left[ {\,\begin{array}{*{20}{c}}1&2\\0&1\end{array}\,} \right] = \left[ {\,\begin{array}{*{20}{c}}1&6\\0&1\end{array}\,} \right]$ and so on.
$\therefore $ ${A^n} = \left[ {\,\begin{array}{*{20}{c}}1&{2n}\\0&1\end{array}\,} \right]$.
Similar Questions
Consider the following information regarding the number of men and women workers in three factories $I,\,II$ and $III$
| Men workers |
Women workers |
|
| $I$ | $30$ | $25$ |
| $II$ | $25$ | $31$ |
| $III$ | $27$ | $26$ |
Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?
A manufacturer produces three products $x,\, y,\, z$ which he sells in two markets. Annual sales are indicated below:
| Market | $x$ | $y$ | $z$ |
| $I$ | $10,000$ | $2,000$ | $18,000$ |
| $II$ | $6,000$ | $20,000$ | $8,000$ |
If unit sale prices of $x, \,y$ and $z$ are Rs. $2.50$, Rs. $1.50$ and Rs. $1.00,$ respectively, find the total revenue in each market with the help of matrix algebra.
