If A is an (n x n)-matrix, under which conditions does the linear system Ax = b have a unique solution? 1-The matrix A is regular. 2-All entries of A are non-zero. 3-The inverse of A exists. 4-rank(A) = n 5-cond(A) = 5 6-A admits a LU-decomposition 7-det(A) = 0 8-A is upper triangular with non-zero diagonal entries. 9-cond(A) = 0
So, I think the solution would be for sure 3 and 4 and not 7.
You are right and also $1$ by definition and $8$ are true indeed for a triangular matrix the determinant is equal to the product of the diagonal entries.