Question

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

Answer 1

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.