<s>[INST] Find the nth power of a matrix Let matrix A is 
$$A=\left (\begin{array}{rrr}
1&0&0 \\
1&1&0\\
0&0&1
\end{array}\right)$$
How can I find the 30 th power of A.. 
Is diagonalization possible?
I found the eigen values But cannot continue. 
 [/INST] Notice that
$$
 \begin{pmatrix}
  1 & 0 \\
  a & 1
 \end{pmatrix}
 \begin{pmatrix}
  1 & 0 \\
  b & 1
 \end{pmatrix}
=
 \begin{pmatrix}
    1 & 0 \\
  a+b & 1
 \end{pmatrix}
 \quad
 \text{for all $a,b \in K$},
$$
where $K$ denotes the ground field. Therefore
$$
 \begin{pmatrix}
  1 & 0 \\
  1 & 1
 \end{pmatrix}^n
=
 \begin{pmatrix}
  1 & 0 \\
  n & 1
 \end{pmatrix}.
$$
Now notice that your matrix is a block matrix of the form
$$
 \begin{pmatrix}
  1 & 0 &   \\
  1 & 1 &   \\
    &   & 1
 \end{pmatrix}
 =
 \begin{pmatrix}
  A &   \\
    & B
 \end{pmatrix}
$$
with
$$
 A =
 \begin{pmatrix}
  1 & 0 \\
  1 & 1
 \end{pmatrix}
 \quad\text{and}\quad
 B = (1).
$$
Therefore
$$
 \begin{pmatrix}
  1 & 0 &   \\
  1 & 1 &   \\
    &   & 1
 \end{pmatrix}^n
=
 \begin{pmatrix}
  A &   \\
    & B
 \end{pmatrix}^n
=
 \begin{pmatrix}
  A^n &   \\
      & B^n
 \end{pmatrix}
=
 \begin{pmatrix}
  1 & 0 &   \\
  n & 1 &   \\
    &   & 1
 \end{pmatrix}.
$$
PS: Alternatively one can realize that your matrix is an elementary matrix, namely the one adding the fist row to the second. Applying this $n$-times is the same as just adding the first row $n$-times to the second, which is represented by the matrix we calculated.
</s>