Question

Let $\mathrm{I}$ be an identity matrix of order $2\times 2$ and $\mathrm{P}=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]$. Then the value of $\mathrm{n}\in \mathrm{N}$ for which${\mathrm{P}}^{\mathrm{n}}=5\mathrm{I}-8\mathrm{P}$ is equal to

Answer 1

Step 1: Given data

The matrix $\mathrm{P}$ is given as:

$\mathrm{P}=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]$

And also the identity matrix of order $2\times 2$ is given as:

$\mathrm{I}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Step 2: Calculate the value of $\mathrm{n}$

Let us suppose $\mathrm{n}=2$. Therefore:

$$\begin{gathered} {\mathrm{P}}^2=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right] \\ \Rightarrow {\mathrm{P}}^2=\left[\begin{array}{cc}-1& 1\\ -5& 4\end{array}\right] \end{gathered}$$

Now let us suppose $\mathrm{n}=4$. Therefore:

$$\begin{gathered} {\mathrm{P}}^4=\left[\begin{array}{cc}-1& 1\\ -5& 4\end{array}\right]\left[\begin{array}{cc}-1& 1\\ -5& 4\end{array}\right] \\ \Rightarrow {\mathrm{P}}^4=\left[\begin{array}{cc}-4& 3\\ -15& 11\end{array}\right] \end{gathered}$$

now let $\mathrm{n}=6$. Therefore,

$$\begin{gathered} {\mathrm{P}}^6={\mathrm{P}}^4{\mathrm{P}}^2 \\ \Rightarrow {\mathrm{P}}^6=\left[\begin{array}{cc}-4& 3\\ -15& 11\end{array}\right]\left[\begin{array}{cc}-1& 1\\ -5& 4\end{array}\right] \\ \Rightarrow {\mathrm{P}}^6=\left[\begin{array}{cc}-11& 8\\ -40& 29\end{array}\right] \end{gathered}$$

Now calculating the value of given expression:

$$\begin{gathered} 5\mathrm{I}-8\mathrm{P}=5\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-8\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right] \\ =\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]-\left[\begin{array}{cc}16& 8\\ 40& -24\end{array}\right] \\ =\left[\begin{array}{cc}-11& -8\\ -40& 29\end{array}\right] \\ ={\mathrm{P}}^6 \end{gathered}$$

Therefore, the value of $\mathrm{n}$ is equal to $6$