For m x n A if Ax b is always consistent what is the rank o

For m x n A, if Ax = b is always consistent, what is the rank of A? Rank of A = What explicitly is Col(A)? Col(A) = Suppose (1 2 3) is in Row(A). Can (1 0 1) be in Null(A)? Why or why not? Consider y\" + ay\' + by = 1, where b notequalto 0. A particular solution can be easily found by inspection; what is it?

Solution

Ax = b is consistent implies there exists the solution

Hence the Rank of A will be min(m,n)

Col(A) = n

The statement is FALSE, since Null(A) can\'t contain the vector (1,0,1)

Since y\'\' + ay\' + by = 1, hence the particular solution will be a constant since the RHS is a constant

Hence the particular solution yp will be 1/b

Since yp\' = 0 and yp\'\' = 0

It also satisfies the equation, hence yp = 1/b