The real rank for C*-algebras was defined by Brown-Pedersen in [1] as a noncommutative analog of covering dimension. Given a unital C*-algebra $A$, its real rank $\mathrm{rr}(A)$ is the smallest natural number $n$ (possibly $n=0$) such that for every $n+1$-tuple $(x_0,\ldots,x_n)$ of selfadjoint elements in $A$ and every $\varepsilon>0$ there exists an $n+1$-tuple $(y_0,\ldots,y_n)$ of selfadjoint elements in $A$ such that $\|x_k-y_k\|<\varepsilon$ for $k=0,\ldots,n$ and such that $\sum_k y_k^*y_k$ is invertible. If no such $n$ exists, one sets $\mathrm{rr}(A)=\infty$. If $A$ is nonunital, one sets $\mathrm{rr}(A)=\mathrm{rr}(\widetilde{A})$ for the minimal unitization $\widetilde{A}$.
Question: Do we have $\mathrm{rr}(M_n(A))\leq\mathrm{rr}(A)$?
It seems that this ought to be true, but I couldn't find such a statement anywhere. For the stable rank (which is defined analogous to the real rank, but using not necessarily selfadjoint tuples) Rieffel proved the formula $\mathrm{sr}(M_n(A))=\lceil \frac{\mathrm{sr}(A)-1}{n}\rceil+1$. Rieffel's method does not seem to generalize to the real rank. In fact, I can show that for the real rank there cannot be a formula computing the real rank of $M_n(A)$ in terms of the real rank of $A$. But one would at least hope for the possiblity to bound $\mathrm{rr}(M_n(A))$ in terms of $\mathrm{rr}(A)$.
[1] Brown, Pedersen. C*-algebras of real rank zero, J. Funct. Anal. 99 (1991)
[2] Rieffel. Dimension and stable rank in the K-theory of C*-algebras, Proc. Lond. Math. Soc. 46 (1983)
