Small entries and a large determinant?

Algebra Level 4

True or False?

There exists a real $8 \times 8$ matrix $A=[a_{ij}]$ such that $|a_{ij}|<1$ for all $i,j$ and $\det A=2018$ .

No Yes not enough information an open problem

1 solution

Otto Bretscher
Dec 16, 2018

First consider an $8\times 8$ Hadamard matrix $H$ , that is, a matrix whose entries are all 1 or -1, with their column vectors being pairwise orthogonal (their dot product is 0). Now $H^T H=8I_8$ , by construction, so that $(\det H)^2=8^8=2^{24}$ and $\det H= \pm 2^{12}=\pm 4096$ . Multiplying a row by $-1$ if necessary, we can assume that $\det H=4096$ . Now $\det(aH)=4096a^8=2018$ for some $a$ with $0<a<1$ , and $A=aH$ is a matrix of the required form. The claim is $\boxed{True}$ .

Very thought-provoking matrix problem, Sir Otto!! Happy Holidays :)

tom engelsman - 2 years, 5 months ago

I'm glad you enjoyed the problem. Happy Holidays to you as well, Sir Tom!

Otto Bretscher - 2 years, 5 months ago

Small entries and a large determinant?

1 solution

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