[Math] Determinant is $1$ if the matrix has its transpose as its inverse. PROOF

determinantlinear algebramatrices

We say that a square matrix $A \in \mathbb{R}^{n\times n}$ is unitary if its inverse is given by its transpose. Show that, for a unitary matrix, one has that $\det A = \pm 1$.

I would like to focus on the info given here =and not drift away into explications that are beyond my level. How, from knowing that the transpose is the inverse can we prove that $|\det A|=1$

I tried the inverse=the transpose with $a, b, c, d$ as my numbers and I tried to match up each term with their corresponding one on the other side and =then solve the equation

Best Answer

Ingredients:

  • What is the definition of inverse? Write it down.
  • $\det(AB)=\det(A)\det(B)$
  • $\det(A)=\det(A^T)$
  • $\det(I)=1$
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