Given matrices $A = \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 4 \\ 1 & 8 \end{bmatrix}$, we need to calculate:

$A^2 + 2AB + B^2.$

Step 1: Calculate $A^2$

$A^2 = A \cdot A = \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix} \cdot \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix}.$ Perform the matrix multiplication: $A^2 = \begin{bmatrix} (2 \cdot 2 + (-1) \cdot 3) & (2 \cdot -1 + (-1) \cdot 0) \\ (3 \cdot 2 + 0 \cdot 3) & (3 \cdot -1 + 0 \cdot 0) \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ 6 & -3 \end{bmatrix}.$

Step 2: Calculate $2AB$

$AB = \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix} \cdot \begin{bmatrix} 5 & 4 \\ 1 & 8 \end{bmatrix}.$ Perform the matrix multiplication: $AB = \begin{bmatrix} (2 \cdot 5 + (-1) \cdot 1) & (2 \cdot 4 + (-1) \cdot 8) \\ (3 \cdot 5 + 0 \cdot 1) & (3 \cdot 4 + 0 \cdot 8) \end{bmatrix} = \begin{bmatrix} 9 & 0 \\ 15 & 12 \end{bmatrix}.$ Now, multiply by 2: $2AB = 2 \cdot \begin{bmatrix} 9 & 0 \\ 15 & 12 \end{bmatrix} = \begin{bmatrix} 18 & 0 \\ 30 & 24 \end{bmatrix}.$

Step 3: Calculate $B^2$

$B^2 = B \cdot B = \begin{bmatrix} 5 & 4 \\ 1 & 8 \end{bmatrix} \cdot \begin{bmatrix} 5 & 4 \\ 1 & 8 \end{bmatrix}.$ Perform the matrix multiplication: $B^2 = \begin{bmatrix} (5 \cdot 5 + 4 \cdot 1) & (5 \cdot 4 + 4 \cdot 8) \\ (1 \cdot 5 + 8 \cdot 1) & (1 \cdot 4 + 8 \cdot 8) \end{bmatrix} = \begin{bmatrix} 29 & 52 \\ 13 & 68 \end{bmatrix}.$

Step 4: Add $A^2 + 2AB + B^2$

$A^2 + 2AB + B^2 = \begin{bmatrix} 1 & -2 \\ 6 & -3 \end{bmatrix} + \begin{bmatrix} 18 & 0 \\ 30 & 24 \end{bmatrix} + \begin{bmatrix} 29 & 52 \\ 13 & 68 \end{bmatrix}.$ Add the matrices element by element: $A^2 + 2AB + B^2 = \begin{bmatrix} 1 + 18 + 29 & -2 + 0 + 52 \\ 6 + 30 + 13 & -3 + 24 + 68 \end{bmatrix} = \begin{bmatrix} 48 & 50 \\ 49 & 89 \end{bmatrix}.$

Final Answer:

$A^2 + 2AB + B^2 = \begin{bmatrix} 48 & 50 \\ 49 & 89 \end{bmatrix}.$

Would you like to go over any specific step in detail?

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Here are five related questions for deeper exploration:

1. How do you verify if the result of matrix multiplication is correct?
2. Can you calculate $(A + B)^2$ and compare it with the above result?
3. What happens when you calculate $A^3 + B^3$?
4. How does the commutative property differ for addition and multiplication of matrices?
5. How do eigenvalues and eigenvectors of $A$ and $B$ relate to their powers?

Tip: Always check the dimensions of the matrices before performing operations to ensure they are compatible.