# In matrix algebra, a matrix K equals $\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&3 \end{pmatrix}$. What is the value of a, if K^7 = $\begin{pmatrix} c&0&0\\ 0&b&0 \\ 0&0&a \end{pmatrix}$?

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In matrix algebra, a matrix K equals $\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&3 \end{pmatrix}$. What is the value of a, if K^7 = $\begin{pmatrix} c&0&0\\ 0&b&0 \\ 0&0&a \end{pmatrix}$?

(a) 2187

(b) 729

(c) 6561

(d) 5^7

I have been asked this question in an online quiz.

The above asked question is from Eigen Value and Time Dependent Problems in chapter Single Variable Problems of Finite Element Method

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Correct option is (a) 2187

Easiest explanation: Since K is a diagonal matrix, its higher powers are obtained by raising its diagonal elements to the same power. If K=$\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&3 \end{pmatrix}$ then K^7=$\begin{pmatrix} 1^7&0&0 \\ 0&1^7&0 \\ 0&0&3^7\end{pmatrix}$. Equating the corresponding elements of $\begin{pmatrix} c&0&0\\ 0&b&0 \\ 0&0&a \end{pmatrix}$ and $\begin{pmatrix} 1^7&0&0 \\ 0&1^7&0 \\ 0&0&3^7\end{pmatrix}$ we get

a=3^7

a=2187.

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