Let A = `[(1,1),(0,1)]` and B = `[(b_(1),b_(2)),(b_(3),b_(4))]` . If `10 A^(10) +Adj (A^(10))` = B then `b_(1)+b_(2)+b_(3)+b_(4)` is equal to :

Options

A. 91
B. 92
C. 111
D. 112

Answer

To solve the problem, we need to find the sum \( b_1 + b_2 + b_3 + b_4 \) given the equation:

\[ 10 A^{10} + \text{Adj}(A^{10}) = B \]

where \( A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \) and \( B = \begin{pmatrix} b_1 & b_2 \\ b_3 & b_4 \end{pmatrix} \).

### Step 1: Calculate \( A^{10} \)

First, we will find \( A^{10} \). To do this, we can observe the pattern in the powers of \( A \).

1. Calculate \( A^2 \):

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

2. Calculate \( A^3 \):

\[ A^3 = A^2 \cdot A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix} \]

3. Continuing this pattern, we can see that:

\[ A^n = \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} \]

Thus, \( A^{10} = \begin{pmatrix} 1 & 10 \\ 0 & 1 \end{pmatrix} \).

### Step 2: Calculate \( \text{Adj}(A^{10}) \)

The adjoint of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:

\[ \text{Adj} = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

For \( A^{10} = \begin{pmatrix} 1 & 10 \\ 0 & 1 \end{pmatrix} \):

\[ \text{Adj}(A^{10}) = \begin{pmatrix} 1 & -10 \\ 0 & 1 \end{pmatrix} \]

### Step 3: Substitute into the equation

Now we substitute \( A^{10} \) and \( \text{Adj}(A^{10}) \) into the equation:

\[ 10 A^{10} = 10 \begin{pmatrix} 1 & 10 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 10 & 100 \\ 0 & 10 \end{pmatrix} \]

Adding the two matrices:

\[ 10 A^{10} + \text{Adj}(A^{10}) = \begin{pmatrix} 10 & 100 \\ 0 & 10 \end{pmatrix} + \begin{pmatrix} 1 & -10 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 10 + 1 & 100 - 10 \\ 0 + 0 & 10 + 1 \end{pmatrix} = \begin{pmatrix} 11 & 90 \\ 0 & 11 \end{pmatrix} \]

### Step 4: Identify elements of matrix \( B \)

From the equation \( 10 A^{10} + \text{Adj}(A^{10}) = B \), we have:

\[ B = \begin{pmatrix} 11 & 90 \\ 0 & 11 \end{pmatrix} \]

Thus, we can identify:

- \( b_1 = 11 \)
- \( b_2 = 90 \)
- \( b_3 = 0 \)
- \( b_4 = 11 \)

### Step 5: Calculate \( b_1 + b_2 + b_3 + b_4 \)

Now, we calculate the sum:

\[ b_1 + b_2 + b_3 + b_4 = 11 + 90 + 0 + 11 = 112 \]

### Final Answer

Thus, the value of \( b_1 + b_2 + b_3 + b_4 \) is:

\[ \boxed{112} \]