Question

If u is a square matrix such that u^tu = i what can we say about det(u)

Answer 1

Recall that


`\det(\mathbf{AB})=\det\mathbf A\det\mathbf B`


`\det\mathbf A=\det\mathbf A^\top`

for any square matrices
`\mathbf A,\mathbf B`. So if


`\mathbf U^\top\mathbf U=\mathbf I`, and
`\det\mathbf I=1`, we have


`\det(\mathbf U^\top\mathbf U)=(\det\mathbf U)^2=1\implies\det\mathbf U=\pm1`