Consider the system of linear equations of n variables Ax b

Consider the system of linear equations of n variables Ax = b. Recall the short hand for the augmented matrix is [A|b]. Prove If Rank(A)

Solution

To illustrate this theorem, let’s look at the simple systems below. x1 + 2x2 = 1 3x1 + x2 = 2 3x1 + 2x2 = 3 6x1 4x2 = 0 3x1 + 2x2 = 3 6x1 4x2 = 6 The augmented matrices for these systems are, respectively, 1 2 1 3 1 2 3 2 3 6 4 0 3 2 3 6 4 6 . Applying the row-reduction algorithm yields the row-reduced form of each of these augmented matrices. The results are, again respectively, 1 0 1 0 1 1 1 2 3 0 0 0 1 1 2 3 1 0 0 0 . From each of these row-reduced versions of the augmented matrices, one can read off the rank of the coefficient matrix as well as the rank of the augmented matrix. Applying Theorem 1.2 to each of these tells us the number of solutions to expect for each of the corresponding systems. We summarize our findings in the table below. System rank[A] rank[A|b] n # of solutions First 2 2 2 1 Second 1 2 2 0 (inconsistent) Third 1 1 2