Question

The equation ax = b is consistent if the augmented matrix [a | b] is:

a) Row equivalent to the identity matrix
b) Row equivalent to a matrix with a row of zeros
c) Row equivalent to a matrix with two rows of zeros
d) Not row equivalent to the identity matrix

Answer 1

The equation ax = b is consistent if the augmented matrix [a | b] can be row reduced to a form with a row of zeros, indicating at least one solution exists for x.

Step-by-step explanation:

row of zeros, indicating at least one The equation ax = b is considered consistent if there exists at least one solution for x. For the augmented matrix [a | b], which represents a system of linear equations, being consistent means it can be row reduced to a form where we can find a solution for x. Therefore, the correct answer is that the equation ax = b is consistent if the augmented matrix [a | b] is) Row equivalent to a matrix with a row of zeroIf a matrix is row equivalent to a matrix with a row of zeros, it indicates that the system is not over-determined and at least one solution exists,

which makes it consistent. Being row equivalent to the identity matrix or having two rows of zeros doesn't guarantee consistency as they might represent different scenarios such as a unique solution or no solution at all.The equation ax = b is consistent if the augmented matrix [a | b] iaRow equivalent to the identity matrixRow equivalent to a matrix with a row of zerocquivalent to a matrix with two rows of zerodNot row equivalent to the identity matrixIn order for the system of equations to be consistent, the augmented matrix must be row equivalent to the identity matrix, which is represented by option (a).