Let `A=[{:(,2,1),(,0,-2):}], B=[{:(,4,1),(,-3,-2):}] and C=[{:(,-3,2),(,-1,4):}]`. Find `A^2+AC-5B`.

Answer

To solve the problem, we need to calculate \( A^2 + AC - 5B \) using the given matrices:

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

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

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

\[ A^2 = A \cdot A = \begin{pmatrix} 2 & 1 \\ 0 & -2 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 0 & -2 \end{pmatrix} \]

Calculating the elements:

- First row, first column: \( 2 \cdot 2 + 1 \cdot 0 = 4 \)
- First row, second column: \( 2 \cdot 1 + 1 \cdot (-2) = 2 - 2 = 0 \)
- Second row, first column: \( 0 \cdot 2 + (-2) \cdot 0 = 0 \)
- Second row, second column: \( 0 \cdot 1 + (-2) \cdot (-2) = 0 + 4 = 4 \)

Thus,

\[ A^2 = \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} \]

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

Next, we calculate \( AC \):

\[ AC = A \cdot C = \begin{pmatrix} 2 & 1 \\ 0 & -2 \end{pmatrix} \cdot \begin{pmatrix} -3 & 2 \\ -1 & 4 \end{pmatrix} \]

Calculating the elements:

- First row, first column: \( 2 \cdot (-3) + 1 \cdot (-1) = -6 - 1 = -7 \)
- First row, second column: \( 2 \cdot 2 + 1 \cdot 4 = 4 + 4 = 8 \)
- Second row, first column: \( 0 \cdot (-3) + (-2) \cdot (-1) = 0 + 2 = 2 \)
- Second row, second column: \( 0 \cdot 2 + (-2) \cdot 4 = 0 - 8 = -8 \)

Thus,

\[ AC = \begin{pmatrix} -7 & 8 \\ 2 & -8 \end{pmatrix} \]

### Step 3: Calculate \( 5B \)

Now, we calculate \( 5B \):

\[ 5B = 5 \cdot \begin{pmatrix} 4 & 1 \\ -3 & -2 \end{pmatrix} = \begin{pmatrix} 20 & 5 \\ -15 & -10 \end{pmatrix} \]

### Step 4: Combine the results to find \( A^2 + AC - 5B \)

Now we substitute the values we found into the expression \( A^2 + AC - 5B \):

\[ A^2 + AC - 5B = \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} + \begin{pmatrix} -7 & 8 \\ 2 & -8 \end{pmatrix} - \begin{pmatrix} 20 & 5 \\ -15 & -10 \end{pmatrix} \]

Calculating the sum and subtraction element-wise:

1. First element: \( 4 - 7 - 20 = 4 - 27 = -23 \)
2. Second element: \( 0 + 8 - 5 = 3 \)
3. Third element: \( 0 + 2 + 15 = 17 \)
4. Fourth element: \( 4 - 8 + 10 = 6 \)

Thus, the final result is:

\[ A^2 + AC - 5B = \begin{pmatrix} -23 & 3 \\ 17 & 6 \end{pmatrix} \]

### Final Answer:
\[ A^2 + AC - 5B = \begin{pmatrix} -23 & 3 \\ 17 & 6 \end{pmatrix} \]