Let $M_{n}$ be the algebra of $n\times n$ complex matrices. By identifying $M_{n}$ with $B(\mathbb{C^{n}})$, the set of all bounded linear maps from the n-dimensional Hilbert space $\mathbb{C^{n}}$ to $\mathbb{C^{n}}$, with operator norm, i.e. $\|x\|=sup_{\eta~\in~\mathbb{C^{n}},~\|\eta\|~\leq1}|x(\eta)|$. It is easy to see that $M_{n}$ is a Banach algebra.
But I do not know how to compute the exact value of norm of a specific $M_{n}$. For instance, if $a=\left(\begin{array}{ccc} 1/2 & 1 & 0 \\ 0 & 1/2 & 0 \\ 0 & 0 & 1/3 \end{array}\right)$ , according to the operator norm defined above, how to compute its norm $\|a\|$?
Martini's answer if perfectly OK.
Alternatively, you can use the fact that for any matrix $M$ we have $$\Vert M\Vert=\sup\{ \sqrt\lambda;\; \lambda\;\hbox{eigenvalue of}\; M^*M\}\, , $$ where $M^*$ is the adjoint matrix.
For this particular example, it should not be too difficult to find the eigenvalues of the $3\times 3$ matrix $M^*M$; but of course this is hopeless for a "general" larger matrix.