Let for `A=[(1,0,0),(2,1,0),(3,2,1)]`, there be three row matrices `R_(1), R_(2)` and `R_(3)`, satifying the relations, `R_(1)A=[(1,0,0)], R_(2)A=[(2,3,0)]` and `R_(3)A=[(2,3,1)]`. If B is square matrix of order 3 with rows `R_(1), R_(2)` and `R_(3)` in order, then
The value of det. `(2A^(100) B^(3)-A^(99) B^(4))` is

Options

A. `-2`
B. `-1`
C. 2
D. -27

Explanation

To solve the given problem, we need to find the value of the determinant \( \det(2A^{100} B^{3} - A^{99} B^{4}) \) where \( A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix} \) and \( B \) is a matrix formed by the row matrices \( R_1, R_2, R_3 \).

### Step 1: Find the determinant of matrix \( A \)

The determinant of matrix \( A \) can be calculated as follows:

\[ \det(A) = 1 \cdot \det\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} - 0 + 0 = 1 \cdot (1 \cdot 1 - 0 \cdot 2) = 1 \]

### Step 2: Define the row matrices \( R_1, R_2, R_3 \)

From the problem, we know:

- \( R_1 A = (1, 0, 0) \)
- \( R_2 A = (2, 3, 0) \)
- \( R_3 A = (2, 3, 1) \)

This implies that \( R_1, R_2, R_3 \) are the rows of matrix \( B \).

### Step 3: Formulate the matrix \( B \)

Let \( B = \begin{pmatrix} R_1 \\ R_2 \\ R_3 \end{pmatrix} \). Since \( R_1, R_2, R_3 \) are defined by their relations with \( A \), we can express \( B \) as:

\[ B = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 2 & 3 & 1 \end{pmatrix} \]

### Step 4: Find the determinant of matrix \( B \)

Using the determinant formula for a \( 3 \times 3 \) matrix:

\[ \det(B) = 1 \cdot \det\begin{pmatrix} 3 & 0 \\ 3 & 1 \end{pmatrix} - 0 + 0 = 1(3 \cdot 1 - 0 \cdot 3) = 3 \]

### Step 5: Use the properties of determinants

We need to calculate \( \det(2A^{100} B^{3} - A^{99} B^{4}) \). We can factor out \( A^{99} \):

\[ \det(2A^{100} B^{3} - A^{99} B^{4}) = \det(A^{99}) \cdot \det(2A B^{3} - B^{4}) \]

Using the property \( \det(A^n) = (\det(A))^n \):

\[ \det(A^{99}) = (\det(A))^{99} = 1^{99} = 1 \]

### Step 6: Calculate \( \det(2A B^{3} - B^{4}) \)

Using the properties of determinants:

\[ \det(2A) = 2^3 \cdot \det(A) = 8 \cdot 1 = 8 \]

Then, we have:

\[ \det(B^{3}) = (\det(B))^3 = 3^3 = 27 \]

Now, we need to compute \( \det(2A B^{3}) \):

\[ \det(2A B^{3}) = \det(2A) \cdot \det(B^{3}) = 8 \cdot 27 = 216 \]

Next, we need to compute \( \det(B^{4}) \):

\[ \det(B^{4}) = (\det(B))^4 = 3^4 = 81 \]

### Step 7: Final determinant calculation

Now we can compute:

\[ \det(2A B^{3} - B^{4}) = \det(2A B^{3}) - \det(B^{4}) = 216 - 81 = 135 \]

### Step 8: Final determinant value

Thus, the final value of the determinant is:

\[ \det(2A^{100} B^{3} - A^{99} B^{4}) = 1 \cdot 135 = 135 \]

### Conclusion

The value of \( \det(2A^{100} B^{3} - A^{99} B^{4}) \) is **135**.