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[121342211]3\left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right]^{3}

The calculator will raise (if possible) the square 33x33 matrix [121342211]\left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right] to the power of 33, with steps shown.
A

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Your Input

Find [121342211]3\left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right]^{3}.

Solution

To raise a matrix to the power of nn, multiply the matrix by itself n−1n - 1 times.

[121342211]2=[121342211]⋅[121342211]=[9116192413795]\left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right]^{2} = \left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right]\cdot \left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right] = \left[\begin{array}{ccc}9 & 11 & 6\\19 & 24 & 13\\7 & 9 & 5\end{array}\right] (for steps, see matrix multiplication calculator).

[121342211]3=[121342211]2⋅[121342211]=[9116192413795]⋅[121342211]=[54683711714780445530]\left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right]^{3} = \left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right]^{2}\cdot \left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right] = \left[\begin{array}{ccc}9 & 11 & 6\\19 & 24 & 13\\7 & 9 & 5\end{array}\right]\cdot \left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right] = \left[\begin{array}{ccc}54 & 68 & 37\\117 & 147 & 80\\44 & 55 & 30\end{array}\right] (for steps, see matrix multiplication calculator).

Answer

[121342211]3=[54683711714780445530]\left[\begin{array}{ccc}1 & 2 & 1\\3 & 4 & 2\\2 & 1 & 1\end{array}\right]^{3} = \left[\begin{array}{ccc}54 & 68 & 37\\117 & 147 & 80\\44 & 55 & 30\end{array}\right]A