We are given that $A$ is a $2 \times 2$ matrix, and we are asked to find $A$ under the condition $A^T + B = C$, where:

$B = \begin{bmatrix} 1 & 3 \\ 1 & 4 \end{bmatrix}, \quad C = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$

Step 1: Start with the equation $A^T + B = C$

This implies:

$A^T = C - B$

Step 2: Compute $C - B$

$C - B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} - \begin{bmatrix} 1 & 3 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} 2 - 1 & 3 - 3 \\ 4 - 1 & 5 - 4 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$

Thus, we have:

$A^T = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$

Step 3: Find $A$

Since $A^T$ is the transpose of $A$, we can obtain $A$ by transposing $A^T$:

$A = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}$

Thus, the matrix $A$ is:

$A = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}$

Let me know if you need more details or have any questions!

Here are 5 related questions to explore further:

1. How do we compute the transpose of a matrix in general?
2. Can matrix addition and subtraction be done with matrices of different dimensions?
3. What properties does a matrix have if it is symmetric?
4. How do you solve a system of equations using matrix operations?
5. What are the properties of matrix multiplication in terms of transpose and inverses?

Tip: Always double-check the dimensions of your matrices before performing operations like addition, subtraction, or multiplication!