If `A=[{:(,1,4),(,2,1):}], B=[{:(,-3,2),(,4,0):}] and C=[{:(,1,0),(,0,2):}]` simplify : `A^2+BC`.

Answer

To simplify the expression \( A^2 + BC \) where

\[ A = \begin{pmatrix} 1 & 4 \\ 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -3 & 2 \\ 4 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}, \]

we will follow these steps:

### Step 1: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:

\[ A^2 = A \times A = \begin{pmatrix} 1 & 4 \\ 2 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 4 \\ 2 & 1 \end{pmatrix} \]

Calculating the elements:

- First row, first column:
\[ 1 \cdot 1 + 4 \cdot 2 = 1 + 8 = 9 \]

- First row, second column:
\[ 1 \cdot 4 + 4 \cdot 1 = 4 + 4 = 8 \]

- Second row, first column:
\[ 2 \cdot 1 + 1 \cdot 2 = 2 + 2 = 4 \]

- Second row, second column:
\[ 2 \cdot 4 + 1 \cdot 1 = 8 + 1 = 9 \]

Thus,

\[ A^2 = \begin{pmatrix} 9 & 8 \\ 4 & 9 \end{pmatrix} \]

### Step 2: Calculate \( BC \)

Next, we calculate the product \( BC \):

\[ BC = B \times C = \begin{pmatrix} -3 & 2 \\ 4 & 0 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} \]

Calculating the elements:

- First row, first column:
\[ -3 \cdot 1 + 2 \cdot 0 = -3 + 0 = -3 \]

- First row, second column:
\[ -3 \cdot 0 + 2 \cdot 2 = 0 + 4 = 4 \]

- Second row, first column:
\[ 4 \cdot 1 + 0 \cdot 0 = 4 + 0 = 4 \]

- Second row, second column:
\[ 4 \cdot 0 + 0 \cdot 2 = 0 + 0 = 0 \]

Thus,

\[ BC = \begin{pmatrix} -3 & 4 \\ 4 & 0 \end{pmatrix} \]

### Step 3: Calculate \( A^2 + BC \)

Now we add the matrices \( A^2 \) and \( BC \):

\[ A^2 + BC = \begin{pmatrix} 9 & 8 \\ 4 & 9 \end{pmatrix} + \begin{pmatrix} -3 & 4 \\ 4 & 0 \end{pmatrix} \]

Calculating the elements:

- First row, first column:
\[ 9 + (-3) = 6 \]

- First row, second column:
\[ 8 + 4 = 12 \]

- Second row, first column:
\[ 4 + 4 = 8 \]

- Second row, second column:
\[ 9 + 0 = 9 \]

Thus,

\[ A^2 + BC = \begin{pmatrix} 6 & 12 \\ 8 & 9 \end{pmatrix} \]

### Final Answer:

\[ A^2 + BC = \begin{pmatrix} 6 & 12 \\ 8 & 9 \end{pmatrix} \]