Finding the 10th root of a matrix

I want to find a $2 \times 2$ matrix, named $A$ in this situation, such that:

$$A^{10}=\begin {bmatrix} 1 & 1 \\ 0 & 1 \end {bmatrix} $$

How can I get started? I was thinking about filling $A$ with arbitrary values $a, b, c, d$ and then multiplying it by itself ten times, then setting those values equal to the given values but I quickly realized that would take too long. Is there a more efficient way?

Solution 1:

Take$$A=\begin{bmatrix}1&x\\0&1\end{bmatrix}.$$Now, compute $A^2,A^3,\ldots$ You'll find quickly which $x$ you should choose.

Solution 2:

You can also consider $$A=\begin {bmatrix} 1 & 0 \\ 0 & 1 \end {bmatrix} + \begin {bmatrix} 0 & a \\ 0 & 0 \end {bmatrix} =I+N $$ and try to use binomial formula for expansion of powers of a binomial.

Notice that $N^k=0$ for $k>1$.