[Math] There are exactly three $2\times 2$ row reduced matrices $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ such that $a+b+c+d=0$

linear algebramatrices

Let $A$ be $2\times 2$ matrix with complex entries,
$$A=\begin{bmatrix} a & b\\ c& d\end{bmatrix}$$
Suppose that $A$ is row reduced and also that $a+b+c+d=0$. Prove that there are exactly three such matrices.

Best Answer

I have seen this question in Section 1.4, Exercise 6 of Linear Algebra by Hoffman & Kunze. I believe both the comment to the original question and the already-existing answer are assuming row-reduced is the same as row-reduced echelon.

In this text, a matrix is row-reduced if:

  1. the first non-zero entry in each non-zero row is equal to 1
  2. each column of the matrix which contains the leading nonzero entry of some row has all its other entries equal to zero.

With this definition, the three matrices are:

$ \begin{bmatrix} 0 & 0\\ 0& 0\end{bmatrix}, \begin{bmatrix} 1 & -1\\ 0& 0\end{bmatrix}, \begin{bmatrix} 0 & 0\\ 1& -1\end{bmatrix} $

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