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A set of linear equations is represented by the matrix equation Ax = b. The necessary condition for the existence of a solution for this system is:
- a)b must be linearly depended on the columns of A
- b)A must be invertible
- c)b must be linearly independent of the columns of A
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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A set of linear equations is represented by the matrix equation Ax = b...
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A set of linear equations is represented by the matrix equation Ax = b...
Necessary Condition for Existence of Solution for Linear Equations
The necessary condition for the existence of a solution for a set of linear equations represented by the matrix equation Ax = b is that A must be invertible.
Explanation:
- Invertible matrix: An invertible matrix is a square matrix that has an inverse, i.e., a matrix that when multiplied with the original matrix gives an identity matrix. A matrix that is not invertible is called a singular matrix.
- Solving linear equations: To solve a set of linear equations, we represent them in matrix form as Ax = b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix. The solution to the system of equations is obtained by finding the inverse of A and multiplying it with b, i.e., x = A^-1b.
- Existence of solution: For a solution to exist, the matrix A must be invertible. If A is not invertible, then the system of equations does not have a unique solution. In other words, there are either no solutions or infinitely many solutions.
- Linearly dependent/independent: The linear dependence or independence of b on the columns of A does not affect the existence of a solution. It only affects the uniqueness of the solution. If b is linearly dependent on the columns of A, then there are infinitely many solutions. If b is linearly independent of the columns of A, then there is a unique solution.
Therefore, the correct answer is option 'A' - A must be invertible.
The necessary condition for the existence of a solution for a set of linear equations represented by the matrix equation Ax = b is that A must be invertible.
Explanation:
- Invertible matrix: An invertible matrix is a square matrix that has an inverse, i.e., a matrix that when multiplied with the original matrix gives an identity matrix. A matrix that is not invertible is called a singular matrix.
- Solving linear equations: To solve a set of linear equations, we represent them in matrix form as Ax = b, where A is the coefficient matrix, x is the variable matrix, and b is the constant matrix. The solution to the system of equations is obtained by finding the inverse of A and multiplying it with b, i.e., x = A^-1b.
- Existence of solution: For a solution to exist, the matrix A must be invertible. If A is not invertible, then the system of equations does not have a unique solution. In other words, there are either no solutions or infinitely many solutions.
- Linearly dependent/independent: The linear dependence or independence of b on the columns of A does not affect the existence of a solution. It only affects the uniqueness of the solution. If b is linearly dependent on the columns of A, then there are infinitely many solutions. If b is linearly independent of the columns of A, then there is a unique solution.
Therefore, the correct answer is option 'A' - A must be invertible.
Community Answer
A set of linear equations is represented by the matrix equation Ax = b...
Ans should be D because if A is invertible then solution exist true but when A is non invertible then also infinte solution can exists when b linearly depends on A. another case if b is linearly independent of A then solution exist when A invertible. So none of given condition are necessary
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A set of linear equations is represented by the matrix equation Ax = b. The necessary condition for the existence of a solution for this system is:a)b must be linearly depended on the columns of Ab)A must be invertiblec)b must be linearly independent of the columns of Ad)None of theseCorrect answer is option 'A'. Can you explain this answer?
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