Two proving problem from the book Algebra by Artin

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SUMMARY

The discussion focuses on two specific problems from the book "Algebra" by Michael Artin. The first problem involves proving that for a 2n x 2n matrix M structured as blocks A, B, C, and D, with A being invertible and AC = CA, the determinant of M equals the determinant of (AD - CB). The second problem requires proving that an n x n matrix A with integer entries has an integer inverse if and only if its determinant is either 1 or -1.

PREREQUISITES
  • Understanding of matrix theory, specifically block matrices.
  • Familiarity with determinants and their properties.
  • Knowledge of invertible matrices and conditions for invertibility.
  • Basic concepts of integer matrices and their inverses.
NEXT STEPS
  • Study block matrix determinants, particularly the formula for 2x2 block matrices.
  • Explore properties of determinants, focusing on integer matrices and their inverses.
  • Review proofs related to the conditions for matrix invertibility.
  • Investigate the implications of determinants being 1 or -1 in the context of integer matrices.
USEFUL FOR

Students of linear algebra, mathematicians interested in matrix theory, and anyone studying proofs in abstract algebra, particularly those using Artin's "Algebra".

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Can anyone give me some hint on proving 1.3 13 and 1.5 3 of Algebra by Artin?
 
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Probably.

It would help if we knew what the questions were. Where are you getting stuck exactly?
 
the first problem : M is a 2n*2n matrix in the form A B C D where each block A(at the position 1,1) B(1,2) C(2,1) D(2,2) is an n*n block. A is invertible and AC=CA. Prove the det M = det (AD - CB)

the second problem:
A is an n*n matrix with integer entries. Prove that the inverse of A has integer entries if and only if det A = 1 or -1
 
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