It is not possible to meet exactly the recommendation by applying some combination of the three(3) brands of fertilizer because having put d question in a Matrix form. The determinant to know whether it is possible or not is equal to Zero(0) which is inconsistent or does not have solution. Therefore the recommendation can not be meet.
It seems like you have described a problem related to solving a system of linear equations using a matrix form. Specifically, you mention that you have a system of equations involving the application of three different brands of fertilizer, and you want to determine if it is possible to meet a certain recommendation.
You mentioned that if you put this system of equations in a matrix form, the determinant of the matrix is equal to zero, indicating that the system is inconsistent and does not have a solution. In other words, there is no combination of the three brands of fertilizer that can meet the recommendation.
To explain further, let's review some concepts. In linear algebra, a determinant is a value that can be calculated for a square matrix. For a 2x2 matrix, the determinant can be calculated as:
| a b |
| c d | = ad - bc
For larger matrices, such as a 3x3 matrix, the determinant can be calculated using a formula called the expansion by minors.
Determinants have various properties, one of which is that if the determinant of a matrix is zero, then the matrix is considered to be singular or non-invertible. This means that the system of equations it represents does not have a unique solution, or it could be inconsistent (i.e., no solution).
In your case, since the determinant of the matrix representing the system of equations is zero, it indicates that the system is inconsistent and cannot be solved. Therefore, it is not possible to meet the recommended values by applying a combination of the three brands of fertilizer.
To verify this result, you could try solving the system of equations using other methods, such as Gaussian elimination or matrix inversion, and you will find that there is no solution.
Remember that determinants can be useful in analyzing systems of equations and understanding whether they have solutions, but they are not the only method for solving such systems. Different methods, such as substitution or elimination, can be used depending on the specific problem.