If A be an n-rowed non-singular matrix, X be an n×1 matrix, B be n×1 matrix, the system of equations AX = B has (CO1)
Understand the Problem
The question is asking about the solutions of a system of equations represented in matrix form (AX = B), where A is a non-singular matrix. It requests to determine the type of solutions (unique, infinite, etc.) based on the properties of the matrix A.
Answer
A
Answer for screen readers
The system of equations ( AX = B ) has a unique solution, so the answer is A.
Steps to Solve
- Identify the properties of matrix A
Since matrix A is specified to be non-singular, it has an inverse and its determinant is non-zero. This indicates that the system of equations represented by ( AX = B ) will have a certain type of solution.
- Determine the implications of a non-singular matrix
For a non-singular matrix ( A ), the system ( AX = B ) will yield exactly one unique solution. This is a fundamental property of non-singular matrices in linear algebra.
- Express the solution relationship
The unique solution can be represented mathematically as:
$$ X = A^{-1}B $$
This equation shows that the solution matrix ( X ) can be calculated by multiplying the inverse of matrix ( A ) with matrix ( B ).
- Conclude on the type of solutions
Since we established that ( A ) is non-singular, we can confidently conclude that ( AX = B ) has a unique solution, which corresponds to option A.
The system of equations ( AX = B ) has a unique solution, so the answer is A.
More Information
In linear algebra, a non-singular matrix (also referred to as invertible) is crucial since it guarantees that a system of linear equations has a unique solution. This is used frequently in solving systems of equations in various applications, including engineering and economics.
Tips
- Confusing non-singular and singular matrices: A singular matrix does not have an inverse and could lead to either no solutions or infinitely many, while a non-singular matrix guarantees a unique solution.
- Ignoring matrix dimensions: Ensure to check that the dimensions of ( A ), ( X ), and ( B ) are compatible to form a proper matrix equation.
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