1. **Problem Statement:** Determine whether the set of all $2 \times 2$ matrices of the form $$\begin{bmatrix} a & a + b \\ a + b & b \end{bmatrix} \quad a,b \in \mathbb{R}$$ with matrix addition and scalar multiplication forms a vector space. List all vector space axioms that fail if any. 2. **Recall Vector Space Axioms:** A set with addition and scalar multiplication is a vector space if it satisfies 8 axioms for addition (closure, associativity, identity, inverse, commutativity) and 4 axioms for scalar multiplication (closure, distributivity over vector addition, distributivity over scalar addition, compatibility with scalar multiplication, and identity scalar multiplication). 3. **Check Closure under Addition:** Take two matrices $$M_1 = \begin{bmatrix} a_1 & a_1 + b_1 \\ a_1 + b_1 & b_1 \end{bmatrix}, \quad M_2 = \begin{bmatrix} a_2 & a_2 + b_2 \\ a_2 + b_2 & b_2 \end{bmatrix}$$ Their sum is $$M_1 + M_2 = \begin{bmatrix} a_1 + a_2 & (a_1 + b_1) + (a_2 + b_2) \\ (a_1 + b_1) + (a_2 + b_2) & b_1 + b_2 \end{bmatrix} = \begin{bmatrix} a' & a' + b' \\ a' + b' & b' \end{bmatrix}$$ where $$a' = a_1 + a_2, \quad b' = b_1 + b_2$$ Check if the $(1,2)$ entry equals $a' + b'$: $$(a_1 + b_1) + (a_2 + b_2) = (a_1 + a_2) + (b_1 + b_2) = a' + b'$$ So closure under addition holds. 4. **Check Closure under Scalar Multiplication:** For scalar $c \in \mathbb{R}$, $$c M_1 = c \begin{bmatrix} a_1 & a_1 + b_1 \\ a_1 + b_1 & b_1 \end{bmatrix} = \begin{bmatrix} c a_1 & c(a_1 + b_1) \\ c(a_1 + b_1) & c b_1 \end{bmatrix}$$ We want to see if this is of the form $$\begin{bmatrix} a'' & a'' + b'' \\ a'' + b'' & b'' \end{bmatrix}$$ for some $a'', b'' \in \mathbb{R}$. Let $$a'' = c a_1, \quad b'' = c b_1$$ Then $$a'' + b'' = c a_1 + c b_1 = c(a_1 + b_1)$$ which matches the $(1,2)$ and $(2,1)$ entries. So closure under scalar multiplication holds. 5. **Check Other Vector Space Axioms:** Since the set is a subset of all $2 \times 2$ matrices and the operations are inherited from matrix addition and scalar multiplication, all axioms except possibly closure hold automatically. 6. **Conclusion:** The set is closed under addition and scalar multiplication and inherits all other vector space axioms from $M_{2 \times 2}(\mathbb{R})$. Therefore, it forms a vector space. **Final answer:** The set of matrices $$\begin{bmatrix} a & a + b \\ a + b & b \end{bmatrix}$$ with $a,b \in \mathbb{R}$ is a vector space under usual matrix addition and scalar multiplication. No axioms fail.