# 2011-11 Skew-symmetric and symmetric matrices

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

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# 2011-5 Linear function on matrices

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.

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# 2010-12 Make a nonsingular matrix by perturbing the diagonal

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.

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# 2010-2 Nonsingular matrix

Let A=(aij) 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.

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# 2008-8 Positive eigenvalues

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

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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.
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.