Answer:
See Below
Step-by-step explanation:
A non-homogeneous equation of the form Ax=b always has a solution IF:
The column space of Matrix A has equal "dimensions" and "column length".
Now,
Given, 12 equations and 13 unknown, so Matrix A can be:
Matrix A = 12 x 13
Hence, the column length = 12
Each column here is a vector in space (vector space).
Now, we essentially need to figure out if the columns span the vector space of .
The rank theorem is:
n is 13 so,
Hence,
Rank A = 12
Hence the dimension is 12 and columns span this.
Thus,
Thus the system always has a solution.