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The determinant function is only defined for an nxn (i.e. square) matrix. So by definition of the determinant it would not exist for a 2x3 matrix.
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No.
A single math equation does not have a determinant. A system of equations (3x3 , 4x4, etc.) will have a determinant. You can find a determinant of a system by converting the system into a corresponding matrix and finding its determinant.
It isn't clear what you want to solve for. If you want to find the matrix, there is not a unique solution - there are infinitely many matrices with the same determinant.