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Here, I will start with a simple expression for an input–output system with $x(t)$ representing the vector of outputs and $A$ the input–output matrix. Then, the simplest possible linear discrete time model of multi-commodity production with no fixed capital, perfect thrift, and no technical change is: $x(t+1) = Ax$. For this system to grow stably over time, there has to be a stable rate of growth a at which all sectors grow: $x(t+1) = (1+\alpha)\cdot x(t)$. These two equations yield $((1+\alpha)\cdot I -A)\cdot x(t) = 0$ - This is only consistent with non-zero output levels if $|(1+\alpha) \cdot I -A|=0$. This is the crucial relationship that basically determines the results to follow, since the stability of a linear difference equation is determined by the dominant eigenvalue of the matrix (the largest root of the polynomial $|(1+\alpha) \cdot I -A|=0$). If this dominant eigenvalue exceeds zero (for a continuous time system) or one (for a discrete time system, such as this example), then the equilibrium of the system will be unstable.

Can anyone explain this - because I can't understand how the whole things are going on. Like why $A$ must have all eigenvalues smaller than 1 to be stable and so on.

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You can find answer to your question on wikipedia:

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