Given matrix `B=[{:(,1,1),(,8,3):}]`. Find the matrix X if, `X=B^2-4B.` Hence, solve for a and b given `X[{:(,a),(,b):}]=[{:(,5),(,50):}]`.

Answer

To solve the problem step-by-step, we will follow the instructions given in the question and perform the necessary calculations.

### Step 1: Find \( B^2 \)

Given the matrix \( B = \begin{pmatrix} 1 & 8 \\ 1 & 3 \end{pmatrix} \), we need to calculate \( B^2 \) by multiplying \( B \) with itself.

\[ B^2 = B \times B = \begin{pmatrix} 1 & 8 \\ 1 & 3 \end{pmatrix} \times \begin{pmatrix} 1 & 8 \\ 1 & 3 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 1 \times 1 + 8 \times 1 = 1 + 8 = 9 \)
- First row, second column: \( 1 \times 8 + 8 \times 3 = 8 + 24 = 32 \)
- Second row, first column: \( 1 \times 1 + 3 \times 1 = 1 + 3 = 4 \)
- Second row, second column: \( 1 \times 8 + 3 \times 3 = 8 + 9 = 17 \)

Thus, we have:
\[ B^2 = \begin{pmatrix} 9 & 32 \\ 4 & 17 \end{pmatrix} \]

### Step 2: Calculate \( 4B \)

Next, we calculate \( 4B \):
\[ 4B = 4 \times \begin{pmatrix} 1 & 8 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 4 \times 1 & 4 \times 8 \\ 4 \times 1 & 4 \times 3 \end{pmatrix} = \begin{pmatrix} 4 & 32 \\ 4 & 12 \end{pmatrix} \]

### Step 3: Calculate \( X = B^2 - 4B \)

Now we can find \( X \):
\[ X = B^2 - 4B = \begin{pmatrix} 9 & 32 \\ 4 & 17 \end{pmatrix} - \begin{pmatrix} 4 & 32 \\ 4 & 12 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 9 - 4 = 5 \)
- First row, second column: \( 32 - 32 = 0 \)
- Second row, first column: \( 4 - 4 = 0 \)
- Second row, second column: \( 17 - 12 = 5 \)

Thus, we have:
\[ X = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} \]

### Step 4: Solve for \( a \) and \( b \)

We are given that:
\[ X \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 5 \\ 50 \end{pmatrix} \]

Substituting \( X \):
\[ \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 5 \\ 50 \end{pmatrix} \]

This results in the following equations:
1. \( 5a = 5 \)
2. \( 5b = 50 \)

From the first equation:
\[ a = \frac{5}{5} = 1 \]

From the second equation:
\[ b = \frac{50}{5} = 10 \]

### Final Answer:
Thus, the values of \( a \) and \( b \) are:
\[ a = 1, \quad b = 10 \]