Let `a_r=r^4C_r,b_r=(4-r)^4C_r,A_r=[{:(a_r,2),(3,b_r):}] and A = sum _(r=0)^4A_r` then the value of |A| is equal to

Answer

To solve the problem step by step, we will first define the components of the matrices and then calculate the determinant of the resulting matrix.

### Step 1: Define \( a_r \) and \( b_r \)
Given:
- \( a_r = \binom{4}{r} \)
- \( b_r = \binom{4}{4-r} = \binom{4}{r} \) (since \( \binom{n}{k} = \binom{n}{n-k} \))

### Step 2: Define the matrix \( A_r \)
The matrix \( A_r \) is defined as:
\[ A_r = \begin{pmatrix} a_r & 2 \\ 3 & b_r \end{pmatrix} \]
Substituting the values of \( a_r \) and \( b_r \):
\[ A_r = \begin{pmatrix} \binom{4}{r} & 2 \\ 3 & \binom{4}{r} \end{pmatrix} \]

### Step 3: Calculate \( A \)
We need to sum \( A_r \) from \( r = 0 \) to \( r = 4 \):
\[ A = \sum_{r=0}^{4} A_r = \sum_{r=0}^{4} \begin{pmatrix} \binom{4}{r} & 2 \\ 3 & \binom{4}{r} \end{pmatrix} \]

### Step 4: Calculate the individual components of the matrix
Calculating each component of the resulting matrix \( A \):
- The (1,1) entry:
\[ \sum_{r=0}^{4} \binom{4}{r} = 2^4 = 16 \]
- The (1,2) entry:
\[ \sum_{r=0}^{4} 2 = 2 \times 5 = 10 \]
- The (2,1) entry:
\[ \sum_{r=0}^{4} 3 = 3 \times 5 = 15 \]
- The (2,2) entry:
\[ \sum_{r=0}^{4} \binom{4}{r} = 2^4 = 16 \]

Thus, we have:
\[ A = \begin{pmatrix} 16 & 10 \\ 15 & 16 \end{pmatrix} \]

### Step 5: Calculate the determinant of matrix \( A \)
The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by \( ad - bc \):
\[ |A| = 16 \cdot 16 - 10 \cdot 15 \]
Calculating this:
\[ |A| = 256 - 150 = 106 \]

### Final Answer
Thus, the value of \( |A| \) is \( 106 \).