Determine whether the statements below are true or false and

Determine whether the statements below are true or false, and justify your answer. A system of four linear equations in three unknowns is always inconsistent. If the matrix A is in reduced row-echelon form, then at least one of the entries in each column of A must be 1. the matrix equation A [x z w] = [0 0 0 1] is inconsistent for all 4 times 3 matrices A.

Solution

False. A system of 4 linear equations in 3 variables is over determined, but not necessarily inconsistent. If one or more of the equations in the system is/are linear combination(s) of the other equations, the system may or may not be inconsistent. False. The RREF of a matrix need not have 1 as an entry in each column. As a matter of fact, a column may have all zeros also. False. The matrix equation AX = ( 0,0,0,1)T , where A is a 4x3 matrix and X = (x,z,w)T need not be always inconsistent. It will be inconsistent only if the last (4th) row of A is a row of zeros.