Linear Algebra: In a test for spanning example in the text, it states that \" th

Question

Linear Algebra: In a test for spanning example in the text, it states that " the system is consistent if and only if its coefficient matrix has a nonzero determinant. But this is not the case here since det(A)=0."

Now, in a solution for a problem in which we must determine if a matrix can be written as a linear combination of 3 given matrices, we are told that since the augmented matrix is consistent, we can conclude that the given matrix is a linear combination. However, the determinant of that augmented matrix is zero. That seems like a contradiction. What am I missing? I understand that if the bottom row has all zeros except for the rightmost term, it is inconsistent because zero can't equal anything but zero, but that's not the issue in the augmented matrix. Am I confused about the definition of consistent or the application of the zero determinant rule or what? Thanks!

Explanation / Answer

I believe you are just confused with the definition consistent system. It is not necessary for the determinant of coefficient matrix to be non-zero for the system to be consistent. That's the condition for system to be consistent and independent. In other words, for a system to have UNIQUE solution, the determinant of the coefficient matrix must be nonzero. Consistent system can have either unique solution or infintely many solutions. That condition which you stated is only applicable for the unique solution case. Hope it clarifies the doubt :)

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