Question

Let A be a 5 x 3 matrix, let y be a vector in R3, and let z be a vector in R5. Suppose Ay = z. What fact allows you to conclude that the system Ax = 5z is consistent?

Answer 1

Step by step approach is as shown

Explanation:

• Consider the system Ax = 5z .............(1)

• Recalling that z = Ay

• Substitute (Ay) for z in equation (1)

• therefore, Ax = 5(Ay) ....................... (2)

• Hence the equation can also be written as Ax = A(5y) ................ (3)

recalling from commutative law that A + B = B + A and since A is a scalar, and from scalar multiplication of matrix.

• From equation (3) ; Ax = A(5y), it implies that x = 5y from comparison and as such if we compare with equation (2) where z = Ay

• therefore equation (2) can then be written as Ax = 5z, since there is consistency as such the the equation will also have a solution.