find formulas for the entries of mn, where n is a positive integer.

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06-12-2022
Mathematics

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find formulas for the entries of mn, where n is a positive integer.

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qwlight qwlight · Answer 1 · 2022-12-12T18:25:20+01:00

The formula for the entries of [tex]M^{n}[/tex], where n is a positive integer is [tex]=\left[\begin{array}{cc}2 \times 9^n-7^n & 7^n-9^n \\2\left(9^n-7^n\right) & 2 \times 7^n-9^n\end{array}\right][/tex] for M = [tex]\left[\begin{array}{cc}11 & -2 \\4 & 5\end{array}\right][/tex] .

Let us find the formula for entries of [tex]M^{n}[/tex], where n is a positive integer.

Let M be a 2 x 2 matrix [tex]\left[\begin{array}{cc}11 & -2 \\4 & 5\end{array}\right][/tex]

we get the eigenvalues and their associated eigenvectors as follows-

Eigenvalue : 9λ , multiplicity : 1λ, eigenvector : [ 1, 1]

Eigenvalue : 7λ , multiplicity : 1λ, eigenvector : [ 0.5, 1]

now, we have

[tex]{\left[\begin{array}{ll}9 & 0 \\0 & 7\end{array}\right] }[/tex] = [tex]\left[\begin{array}{cc}1 & \frac{1}{2} \\1 & 1\end{array}\right]^{-1}[/tex] [tex]\left[\begin{array}{cc}11 & -2 \\4 & 5\end{array}\right][/tex] [tex]\left[\begin{array}{cc}1 & \frac{1}{2} \\1 & 1\end{array}\right][/tex]

= [tex]\left[\begin{array}{cc}2 & -1 \\-2 & 2\end{array}\right][/tex] [tex]\left[\begin{array}{cc}11 & -2 \\4 & 5\end{array}\right][/tex] [tex]\left[\begin{array}{cc}1 & \frac{1}{2} \\1 & 1\end{array}\right][/tex]

hence, [tex]\left[\begin{array}{cc}11 & -2 \\4 & 5\end{array}\right][/tex] = [tex]\left[\begin{array}{cc}1 & \frac{1}{2} \\1 & 1\end{array}\right][/tex] [tex]\left[\begin{array}{cc}9 & 0 \\0 & 7\end{array}\right][/tex] [tex]\left[\begin{array}{cc}2 & -1 \\-2 & 2\end{array}\right][/tex]

So,

[tex]{\left[\begin{array}{cc}11 & -2 \\4 & 5\end{array}\right]^n } & =\left[\begin{array}{ll}1 & \frac{1}{2} \\1 & 1\end{array}\right]\left[\begin{array}{ll}9 & 0 \\0 & 7\end{array}\right]^n\left[\begin{array}{cc}2 & -1 \\-2 & 2\end{array}\right][/tex]

[tex]= \left[\begin{array}{ll}1 & \frac{1}{2} \\1 & 1\end{array}\right]\left[\begin{array}{cc}9^n & 0 \\0 & 7^n\end{array}\right]\left[\begin{array}{cc}2 & -1 \\-2 & 2\end{array}\right][/tex]

[tex]=\left[\begin{array}{cc}2 \times 9^n-7^n & 7^n-9^n \\2\left(9^n-7^n\right) & 2 \times 7^n-9^n\end{array}\right][/tex]

Thus, the formula for the entries of [tex]M^{n}[/tex], where n is a positive integer is [tex]=\left[\begin{array}{cc}2 \times 9^n-7^n & 7^n-9^n \\2\left(9^n-7^n\right) & 2 \times 7^n-9^n\end{array}\right][/tex] for M = [tex]\left[\begin{array}{cc}11 & -2 \\4 & 5\end{array}\right][/tex] .