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LA is a simple and natural logical system for reasoning about matrices. We show that LA, over finite fields, proves a host of matrix identities (so called "hard matrix identities") from the matrix form of the pigeonhole principle. LAP is LA with matrix powering; we show that LAP extended with quantification over permutations is strong enough to prove fundamental theorems of linear algebra (such as the Cayley-Hamilton Theorem). Furthermore, we show that LA with quantification over permutationsdoi:10.1016/j.tcs.2005.09.021 fatcat:f33uoja7bzgnxngj5rlhpdedoe