For blank one a solution or no solution Blank two every or

For blank one - a solution or no solution
Blank two - every or some

Blank three - true or false

Blank four - a nontrivial or no

Solution

Note: The equation Ax=0 is always consistent because the trivial solution i.e. x=0 is always a solution.

Here, it is given that for a nXn matrix A, the equation Ax=b is inconsistent for some b in Rⁿ . It implies that rank(A) < n. Now, in equation Ax=0, since rank(A) < n, therefore this equation has infinitely many solutions.

The statement Ax=b is inconsistent for some b is equivalent to the statement that Ax=b has no solution for some b. From this, all of the statements in the invertible matrix theoremare false including the statement that Ax=0 has only the trivial solution. Thus, Ax=0 has a non-trivial solution.