[Math] Does the set of all 3×3 echelon form matrices with elements in R form a subspace of M3x3(R)? Same question for reduced echelon form matrices.

Question

Screenshot of the past exam question

Firstly, the zero 3×3 matrix denoted as A is both in echelon and reduced echelon form since it satisfies both definitions respectively.

$$
A = \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}
$$

Question 1: Are 3×3 echelon form matrices closed under addition and scalar multiplication?

Question 2: Are 3×3 reduced echelon form matrices closed under addition and scalar multiplication?

Thanks!

Answer 1

Reduced echelon form matrices most certainly do not form a vector space, as $ I_3 $ is in reduced echelon form, and yet $ 2 I_3 = I_3 + I_3$ is not, so it is closed under neither addition nor scalar multiplication.

The set of matrices in echelon form do not form a vector space either: consider

$$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} $$

The third column contains a leading $ 1 $, and yet it has other nonzero entries, which means that it is not in echelon form, and yet the summands are.