Answer: We can see that the coefficient matrix [2 3; 4 6] is non-invertible since the second row is a multiple of the first row. This implies that the equations are dependent, and there are infinitely many solutions that satisfy the system. In this case, any values of x and y that satisfy the equation 2x + 3y = 6 will also satisfy the equation 4x + 6y = 12.
In summary, a non-invertible coefficient matrix in a system of equations indicates the absence of a unique solution. It can result in either no solution due to inconsistent or contradictory equations or infinitely many solutions due to dependent or overlapping equations. Understanding the implications of a non-invertible coefficient matrix helps in analyzing and solving systems of equations effectively.
Explanation:
When the coefficient matrix in a system of equations is non-invertible, it has important implications for the solutions of the system. The coefficient matrix represents the relationship between the variables in the system, and its invertibility determines whether a unique solution exists or not.
Here are the potential scenarios that may arise when dealing with a non-invertible coefficient matrix:
1. No Solution: If the coefficient matrix is non-invertible, it means that the system of equations does not have a unique solution. This can occur when the equations are inconsistent or contradictory. Inconsistent equations lead to a system that has no solution, meaning that the set of equations cannot be satisfied simultaneously. Contradictory equations result in a system where the equations are mutually exclusive, making it impossible to find a common solution.
2. Infinitely Many Solutions: In some cases, a non-invertible coefficient matrix can lead to infinitely many solutions. This occurs when the equations are dependent or overlapping. Dependent equations imply that one equation can be derived from the other equations in the system. As a result, there are an infinite number of solutions that satisfy the equations because they are not uniquely determined.
To better understand these scenarios, let's consider an example:
2x + 3y = 6
4x + 6y = 12
If we write this system of equations in matrix form:
[2 3] [x] [6]
[4 6] * [y] = [12]