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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

1 Answer

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by (185k points)
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Best answer
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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