Can you use Cramer’s rule to solve a linear system with 2×3 coefficient matrix? Explain.
The Correct Answer and Explanation is:
Cramer’s Rule cannot be used to solve a linear system with a 2×3 coefficient matrix. Cramer’s Rule applies specifically to square systems of linear equations, meaning the number of equations must equal the number of unknowns. In other words, for Cramer’s Rule to work, the system must have a square coefficient matrix (n x n), where “n” represents the number of variables or unknowns in the system.
For example, if you have a system of two equations and two unknowns, you can apply Cramer’s Rule because the coefficient matrix would be 2×2. However, if you have a system with two equations and three unknowns (a 2×3 matrix), the system is overdetermined and will not yield a unique solution because there are fewer equations than unknowns.
In a 2×3 system, you cannot directly find the determinant of the coefficient matrix (since it’s not square). For Cramer’s Rule to work, you must first have a square matrix, and if the coefficient matrix is not square (like in the 2×3 case), you cannot proceed with the method.
To solve a 2×3 system, you would typically use Gaussian elimination, matrix inversion (if the system is overdetermined but consistent), or other methods like least squares (if there is no exact solution and you need the best approximation). The system might also be inconsistent or have infinitely many solutions depending on the relationships between the equations.
In summary, Cramer’s Rule is not applicable to non-square systems, such as a 2×3 system, and alternative methods must be employed to solve it.
