Q1: Which equation is correct for the matrix $A = \begin{pmatrix} 2 & 9 \ 9 & -2 \end{pmatrix}$ and the standard inner product?

A1: $ \langle \mathbf{y}, A \mathbf{x} \rangle = \langle A \mathbf{y}, \mathbf{x} \rangle $ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n $.

A2: $ \langle \mathbf{y}, A \mathbf{x} \rangle = \langle \mathbf{x}, A \mathbf{y} \rangle $ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n $.

A3: $ \langle \mathbf{y}, \mathbf{x} \rangle = \langle \mathbf{x}, A^{-1} \mathbf{y} \rangle $ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n $.

A4: $ \langle \mathbf{y}, \mathbf{x} \rangle = \langle \mathbf{x}, A^T \mathbf{y} \rangle $ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n $.

Q2: Let $A$ be a matrix and $A^T$ be the transpose. What is correct for the standard inner product?

A1: $ \langle \mathbf{y}, A \mathbf{x} \rangle = \mathbf{y}^T A \mathbf{x}$ for all $\mathbf{x}, \mathbf{y} $.

A2: $ \langle \mathbf{y}, A \mathbf{x} \rangle = \mathbf{y} A^T \mathbf{x}$ for all $\mathbf{x}, \mathbf{y} $.

A3: $ \langle \mathbf{y}, A \mathbf{x} \rangle = \mathbf{y} A^T \mathbf{x}^T$ for all $\mathbf{x}, \mathbf{y} $.

A4: $ \langle \mathbf{y}, A \mathbf{x} \rangle = \mathbf{y} A \mathbf{x}^T$ for all $\mathbf{x}, \mathbf{y} $.

Q3: What is an alternative definition for the transpose of $A$?

A1: $A^T$ is the only matrix $B$ that satisfies $\langle \mathbf{y}, A \mathbf{x} \rangle = \langle B \mathbf{y}, \mathbf{x} \rangle$.

A2: $A^T$ is the only matrix $B$ that satisfies $\langle \mathbf{y}, A \mathbf{x} \rangle = \langle B^{-1} \mathbf{y}, \mathbf{x} \rangle$.

A3: $A^T$ is the only matrix $B$ that satisfies $\langle \mathbf{y}, A \mathbf{x} \rangle = \langle \mathbf{y}, B \mathbf{x} \rangle$.