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?

Question

Mathematics

Rayne Hardy

25 November, 05:03

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?

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Answers (1)

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Woodard · Answer 1

Step by step approach is as shown

Step-by-step 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.