Prove that for every skew-symmetric matrix A, there are symmetric matrices B and C such that A=BC-CB.

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Prove that for every skew-symmetric matrix A, there are symmetric matrices B and C such that A=BC-CB.

Find all linear functions *f* on the set of n×n matrices such that *f*(XY)=*f*(YX) for every pair of n×n matrices X and Y.

Added: The value f(X) is a scalar.

Let A be a square matrix. Prove that there exists a diagonal matrix J such that A+J is invertible and each diagonal entry of J is ±1.

Let A=(a_{ij}) be an n×n matrix of complex numbers such that \(\displaystyle\sum_{j=1}^n |a_{ij}|<1\) for each i. Prove that I-A is nonsingular.

Let A be a 0-1 square matrix. If all eigenvalues of A are real positive, then those eigenvalues are all equal to 1.

Let A, B be \(3\times 3\) integer matrices such that A, A+B, A+2B, A+3B, A-B, A-2B, A-3B are invertible and their inverse matrices are all integer matrices.

Prove that A+4B also has an inverse, and its inverse is again an integer matrix.

A, B가 \(3\times 3\) 정수 행렬이면서, A, A+B, A+2B, A+3B, A-B, A-2B, A-3B가 모두 역행렬을 가지고 그 역행렬이 모두 정수행렬이라고 하자. 이때 A+4B 역시 역행렬을 가지고 그 역행렬은 정수행렬임을 보여라.