Respuesta :
A 3 × 3 matrix has three rows and three columns, while a zero row is a row
that has all the elements as zero
The correct response option is as follows;
No. Since Ax = y has no solution, then A cannot have a pivot in every row.
Since A is a 3 × 3, it has at most two pivot positions. So the equation Ax = z
has at most two basic variables and at least one free variable for any z.
Thus there is no solution for Ax = z
Reason:
There are no free variables if Ax = z has a solution that is unique, given that
all the rows would have a pivot. However, there exist a zero row '0 0 0' in
the row reduced Echelon form of the coefficient matrix A, given that Ax = y
has no solution. and the matrix A cannot have a pivot at every row [0 0 0 1]
Also there would be no free columns in the row reduced matrix A where Ax
= z has a solution that is unique, because it will result in [Az] having free
variables, which is contradicting the fact that matrix A has a free column
Therefore, a vector y ∈ R³ such that Ax = y has a unique solution does not exist
Therefore;
No. Since Ax = y has no solution, then A cannot have a pivot in every row.
Since A is a 3 × 3, it has at most two pivot positions. So the equation Ax = z
has at most two basic variables and at least one free variable for any z.
Thus there is no solution for Ax = z
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