Chapter 2 - Matrices and Systems of Linear Equations - 2.2 Matrix Algebra - Problems - Page 137: 31

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If $A=xI_n$ and $A=n \times n$ matrix with trace $k$, then $x.n=tr(A)=k$. We get $x=k/n$ from here. Since $n$ is an arbitrary positive integer, there is a unique scalar matrix $k/n$ with a given constant $k$ as its trace.