Question

Which of the following statements is true about the solution to the system of linear equations ax = b, where a is an m-by-n matrix? a. The solution always exists, regardless of the values of a and b. b. The solution depends on the values of m and n. c. The solution is unique for all values of a and b. d. The solution can only be found using the Gauss-Jordan elimination method.

Answer 1

The solution to the system of linear equations (ax = b) may not always exist, contingent on the specific values of (a) and (b).

Step-by-step explanation

The existence of a solution to the system (ax = b) is contingent upon the properties of the matrix (a) and the vector (b). It's incorrect to assert that a solution invariably exists, regardless of the values of (a) and (b) (option a). Rather, it's the characteristics of (a) and (b) that determine the existence and uniqueness of a solution.

The solution's dependency on the values of (m) and (n) (option b) oversimplifies the matter. The critical factor is the nature of the matrix (a) (its rank, invertibility, etc.) and the relationship between (a) and (b) that dictate solution existence and uniqueness.

For instance, if (a) is a square matrix (same number of rows and columns), and it's invertible, a unique solution exists for any non-zero vector (b).

However, if (a) is not square or not invertible, the existence and uniqueness of a solution become more complex and may rely on the specifics of (b). Moreover, the solution isn't exclusively determined using Gauss-Jordan elimination (option d); various methods exist depending on matrix properties.

Answer 2

The correct statement is: b. The solution depends on the values of m and n.

In a system of linear equations represented by the matrix equation
`\(ax = b\)`, where a is an
`\(m\)-by-\(n\)` matrix, the existence and uniqueness of the solution depend on the specific properties of the matrix
`\(a\)` and the vector
`\(b\)`. If the system is consistent (has at least one solution) and the number of equations (rows in a) is equal to the number of unknowns (columns in a), then there is a unique solution. Otherwise, there may be no solution or infinitely many solutions.

The statement that the solution depends on the values of
`\(m\) and \(n\)` reflects this variability in possible solutions based on the dimensions of the matrix a.