I need help with this one question I found from my Linear Algebra textbook. While I was doing problems and checking my work from the back of my textbook, I found out one of the problems I did is wrong. The question was to find the tr(A). The book says the answer should be undefined but it doesn't give any explanation. I got an answer as 3+2 = 5 (Adding numbers diagonally from the matrix).
On
The answer isn't "The formula doesn't work, because of this and that reason". The formula does work (at least once you specify that we are working with the main diagonal). Your example shows this.
The answer is "We just don't do traces of non-square matrices". The trace isn't just a formula. It has a number of useful properties that help us reason about matrices. And many of those properties are mostly only relevant for square matrices.
In fact, one may argue that the formula is one of these secondary properties, and the trace is defined through some other fundamental means, like "the trace is the sum of the eigenvalues". Therefore the trace is, by convention, only defined for square matrices.
The trace of a square matrix is defined as the sum of the entries of the main diagonal and undefined for non-square matrices. And your matrix is not a square matrix.