there is an orthogonal matrices nxn,in which all of its members are
whole numbers (like -1 ,-7,4,8,..).
also there is vector b its members are whole too.
prove that for the system Ax=b we have a single solution "c"
where c=(c1,..,cn) and c1 .. cn are whole numbers
tonio, i think that invoking the determinant is misleading. orthogonal matrices exist that have non-integral entries, example:
[ 1/√2 -1/√2].
more pertinent is the fact that for an orthogonal matrix A, A^-1 = A^T, and clearly A^T has integer entries iff A does.
asking, relations between the transpose of a matrix and its inverse and , which usually are studied
within the frame of vector spaces with inner product, could still be far away from him.
sure, i understand, but how do you go from that fact that det(A) = ±1, to conclude that A^-1 has integral entries?
i exhibited an orthogonal matrix for which this is NOT true. in other words, i am wondering what YOU were thinking as the answer
to the question: "Why (does A^-1 have integer entries)?. What's the determinant of an orthogonal matrix?".
because if your answer is "because det(A) = ±1", that's not correct. having a determinant of ±1, does NOT imply integral entries.