Find the matrix A, if `B=[{:(,2,1),(,0,1):}] and B^2=B+1/2A`.

Answer

To find the matrix \( A \) given the matrix \( B \) and the equation \( B^2 = B + \frac{1}{2} A \), we will follow these steps:

### Step 1: Write down the matrix \( B \)
The matrix \( B \) is given as:
\[ B = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} \]

### Step 2: Calculate \( B^2 \)
To find \( B^2 \), we multiply matrix \( B \) by itself:
\[ B^2 = B \times B = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} \]

Using the matrix multiplication rules:
- First row, first column: \( 2 \times 2 + 1 \times 0 = 4 \)
- First row, second column: \( 2 \times 1 + 1 \times 1 = 2 + 1 = 3 \)
- Second row, first column: \( 0 \times 2 + 1 \times 0 = 0 \)
- Second row, second column: \( 0 \times 1 + 1 \times 1 = 0 + 1 = 1 \)

Thus, we have:
\[ B^2 = \begin{pmatrix} 4 & 3 \\ 0 & 1 \end{pmatrix} \]

### Step 3: Substitute \( B^2 \) into the equation
We substitute \( B^2 \) into the equation \( B^2 = B + \frac{1}{2} A \):
\[ \begin{pmatrix} 4 & 3 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} + \frac{1}{2} A \]

### Step 4: Rearrange the equation to solve for \( A \)
Rearranging gives us:
\[ \frac{1}{2} A = B^2 - B \]
Thus,
\[ A = 2(B^2 - B) \]

### Step 5: Calculate \( B^2 - B \)
Now we calculate \( B^2 - B \):
\[ B^2 - B = \begin{pmatrix} 4 & 3 \\ 0 & 1 \end{pmatrix} - \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix} \]
Calculating the elements:
- First row, first column: \( 4 - 2 = 2 \)
- First row, second column: \( 3 - 1 = 2 \)
- Second row, first column: \( 0 - 0 = 0 \)
- Second row, second column: \( 1 - 1 = 0 \)

Thus,
\[ B^2 - B = \begin{pmatrix} 2 & 2 \\ 0 & 0 \end{pmatrix} \]

### Step 6: Multiply by 2 to find \( A \)
Now we multiply the result by 2:
\[ A = 2 \times \begin{pmatrix} 2 & 2 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 0 & 0 \end{pmatrix} \]

### Final Answer
The matrix \( A \) is:
\[ A = \begin{pmatrix} 4 & 4 \\ 0 & 0 \end{pmatrix} \]