Kunze Linear Algebra · Problem 1.3.6

A row-reduced $2 \times 2$ matrix has a very specific structure: it's either the identity, a single pivot in the first column, or a single pivot in the second column. The constraint $a + b + c + d = 0$ is a linear equation that eliminates most possibilities—you need to systematically check which row-reduced forms can satisfy it. The key is recognizing that the sum of all entries being zero severely restricts the free parameters available in each case.

A row-reduced $2 \times 2$ matrix has a very specific structure: it's either the identity, a single pivot in the first column, or a single pivot in the second column. The constraint $a + b + c + d = 0$ is a linear equation that eliminates most possibilities—you need to systematically check which row-reduced forms can satisfy it. The key is recognizing that the sum of all entries being zero severely restricts the free parameters available in each case.