2014-05 Nonnegative determinant

Let \(n\), \(k\) be positive integers and let \(A_1,A_2,\ldots,A_n\) be \(k\times k\) real matrices. Prove or disprove that \[ \det\left(\sum_{i=1}^n A_i^t A_i\right)\ge 0.\]  (Here, \(A^t\) denotes the transpose of the matrix \(A\).)

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