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Matrices as a sum of two squares matrices

Source: Brazil Undergrad MO 2020 #4

March 18, 2021
linear algebracollege contestBrazilian Undergrad MO 2020Matrix algebra

Problem Statement

For each of the following, provide proof or a counterexample:
a) Every 2×22\times2 matrix with real entries can we written as the sum of the squares of two 2×22\times2 matrices with real entries. b) Every 3×33\times3 matrix with real entries can we written as the sum of the squares of two 3×33\times3 matrices with real entries.
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