Compute the limit of a matrix We need to compute the limit of a sequence as $x \rightarrow \infty$
Using matrix-matrix multiplication we can define power as $A^p=A*\cdots*A$, $p$ times.
We need to compute the limit of $A^p$ as $p \rightarrow \infty$
Now my question is, is it fine to compute some of them and telling that it will approach $0$ or not?
Like for example starting from computing $A^2$, then $A^4$, $\ldots,A^{20}$.
Is this good for showing a limit or not?
 A: For this we can use the Jordan Normal Form. Any generalized eigenspace with generalized eigenvalue which has a modulus (absolute value) smaller than 1 will go to 0, since moduluses multiply.
The Jordan blocks are upper triangular matrices with (same) lambda along the diagonal and 1 along the first off-diagonal. Because of the block multiplication property, values along the diagonal will be raised to the same power as the matrix. And triangular matrices always have their (generalized) eigenvalues along the diagonal. Thus powers of matrices carry over to their (generalized) eigenvalues. So if one eigenvalue is "smaller than 1", it will keep shrinking the larger the exponent becomes.
A: For any multiplicative matrix norm $\|\cdot\|$, $A^p$ will approach zero if and only if there is a $p \geq 0$ such that $\|A^p\| < 1$.  Intuitively, this means that if the sequence $\{A^p\}$ gets "close enough" to $0$, it has to converge to $0$.
An example of such a matrix norm is the $1$-norm.  In particular, if $A$ has entries $a_{ij}$, then
$$
\|A\|_1 = \max_{j=1,\dots,n}\sum_{i=1}^n |a_{ij}| 
$$
If $\|A^p\|_1 < 1$ for any $p \geq 0$, then we can guarantee that $A^p \to 0$ as $p \to \infty$.  
However, given that $A^p \to 0$, we have no guarantee as to how high $p$ would have to be for $\|A^p\|$ to dip below $1$. 
