$F_1$ and $F_2$ be two finite dimensional ${\mathrm{C}}^*$-algebras. Suppose that there are two unital homomorphisms $\varphi_0,\varphi_1\,:F_1\rightarrow F_2$. Denote $$A=A(F_1,F_2,\varphi_0,\varphi_1)=\{(f,a)\in C([0,1],F_2)\oplus F_1 :f(0)=\varphi_0(a)\,\,and\,\,f(1)=\varphi_1(a)\}.$$

These ${\mathrm{C}}^*$-algebras have been introduced into the Elliott program by Elliott and Thomsen. Denote by $\mathcal{C}$ the class of all unital ${\mathrm{C}}^*$-algebras of the form $A(F_1,F_2,\varphi_0,\varphi_1)$ and all the finite dimensional ${\mathrm{C}}^*$-algebras. These ${\mathrm{C}}^*$-algebras will be called Elliott-Thomsen building blocks. A unital ${\mathrm C}^*$-algebra $A\in\mathcal{C}$ is minimal, or a minimal block, if it is indecomposable, i.e., not the direct sum of two or more ${\mathrm C}^*$-algebras in $\mathcal{C}$(see also [@GLN]).

Let $A=A(F_1,F_2,\varphi_0,\varphi_1)$, where $F_1=\oplus_{j=1}^p M_{k_j}(\mathbb{C}),\,F_2=\oplus_{i=1}^lM_{l_i}(\mathbb{C})$ and $\varphi_0,\varphi_1 :F_1\rightarrow F_2$ be unital homomorphisms. Let $\varphi_{0*},\varphi_{1*}:K_0(F_1)=\mathbb{Z}^p\rightarrow K_0(F_1)=\mathbb{Z}^l$ be represented by matrices $\alpha=({\alpha_{ij}})_{l\times p}$ and $\beta=({\beta_{ij}})_{l\times p}$, where $\alpha_{ij},\beta_{ij}\in \mathbb{Z}_+$ for each pair $i,j$. Then $$K_0(A)=
  \left\{
  \left(
\begin{array}{c}
  x_1 \\
  x_2 \\
   \vdots\\
  x_p
\end{array}
  \right)
\in \mathbb{Z}^p,\,\,with\,\,(\alpha -\beta)
  \left(
\begin{array}{c}
  x_1 \\
  x_2 \\
   \vdots\\
  x_p
\end{array}
  \right)
  =0
\right\}$$ $$K_1(A)=\mathbb{Z}^l/Im(\alpha-\beta).$$

We use the notation $\#(\cdot)$ to denote the cardinal number of the set, the sets under consideration will be sets with multiplicity, and then we shall also count multiplicity when we use the notation $\#$.

Let $\phi:A\rightarrow M_n(\mathbb{C})$ be a homomorphism. Then there exists a unitary $u$ such 