Any system of linear equations is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. The highest possible rank of augmented matrix is 3 and is greater than the rank of A. Hence highest possible rank of A is 2.

Rank of Matrix A in an Inconsistent System of Equations

The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. In an inconsistent system of equations, there are no solutions that satisfy all the equations simultaneously. This means that the system is either overdetermined (more equations than variables) or inconsistent (contradictory equations).

To determine the highest possible rank of matrix A in an inconsistent system of equations, we need to analyze the relationship between the number of equations and variables.

Number of Equations and Variables
In this case, matrix A is a 3x4 real matrix, which means it has 3 rows and 4 columns. Let's denote the number of equations as m and the number of variables as n.

- Number of equations (m) = 3
- Number of variables (n) = 4

Rank-Nullity Theorem
According to the Rank-Nullity theorem, for any matrix A, the sum of the rank and nullity is equal to the number of columns in A. Mathematically, it can be represented as:

Rank(A) + Nullity(A) = n

Where n is the number of columns in matrix A.

Rank and Nullity in an Inconsistent System
In an inconsistent system of equations, there are no solutions, which means the nullity of matrix A is zero. Therefore, we can rewrite the Rank-Nullity theorem as:

Rank(A) + 0 = n
Rank(A) = n

This implies that the highest possible rank of matrix A in an inconsistent system is equal to the number of variables (n). In this case, the highest possible rank of matrix A is 4.

Conclusion
In summary, the highest possible rank of matrix A in an inconsistent system of equations is equal to the number of variables. In this case, the rank of matrix A is 4, given that it is a 3x4 real matrix. The rank represents the maximum number of linearly independent rows or columns in the matrix.