Statement: If det() = 7, then = 0 has only the trivial solution
We can prove that the matrix A is invertible if Ax=0 has only a trivial solution in terms of: "Invertible only if the trivial solution is the only element of the null space" by utilizing the rank-nullity theorem.
If A is invertible, it is full rank Rank(A) + Nullity(A) = dim A The null space is the set of vectors x s.t. Ax=0
Now, how can I apply the same logic in terms of the statement above, and how can I prove if it is indeed true or false?