An exercise applying the Cayley-Hamilton theorem

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An exercise applying the Cayley-Hamilton theorem

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#1 h Mar 29, 2025, 4:43 PM

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Let $A = (a_{ij})$ be a nonzero square matrix of order $n$ satisfying
$a_{ik} a_{jk} = a_{kk} a_{ij}, \quad \text{for all } i, j, k.$ Denote by $\operatorname{tr}(A)$ the trace of $A$ , which is the sum of the diagonal elements of $A$ .

a) Prove that $\operatorname{tr}(A) \neq 0$ .

b) Compute the characteristic polynomial of $A$ in terms of $\operatorname{tr}(A)$ .

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#2 h Yesterday at 8:59 AM • 1 Y

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Choose $i,k$ with $a_{ik}\not=0$ . Taking $i=j$ we find $a_{ik}^2=a_{kk}a_{ii}\not=0$ hence $a_{kk}\not=0$ in particular.
Since $a_{ij}={{a_{ik}a_{jk}}\over {a_{kk}}}$ we see that $A$ is symmetrical. Also all elements can be expressed in the $k$ -th column.
We have $a_{il}={{a_{lk}}\over {a_{kk}}}\cdot a_{il}$ hence the $l$ -th column is a multiple of the $k$ th column and $A$ has rank 1.
Then $A$ has the form $Ax=(x,v)w$ for some nonzero vectors $v,w$ .
We have an eigenvector $w$ with eigenvalue $\lambda$ (of alg mpl 1) and all other eigenvalues are 0.
Hence $\lambda={\rm Tr}(A)$ and $F_A=t^{n-1}(t-\lambda)$ .

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