A is a square matrix of order 3 and `v_1, v_2, v_3` are 3 column matrices such that `Av_1=[[1],[2],[3]]`, `Av_2=[[-1],[0],[2]]`, `Av_3=[[0],[-1],[2]]` where `v_1=[[1], [1], [1]`, `v_2=[[2], [0], [3]`, `v_3=[[1], [1], [-1]` then the value of `|A|` is

Options

A. 9
B. `9/2`
C. `9/4`
D. `9/8`

Answer

To find the value of the determinant of the matrix \( A \), we can follow these steps:

### Step 1: Set Up the Matrix Equation
Given that:
\[ Av_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad Av_2 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix}, \quad Av_3 = \begin{bmatrix} 0 \\ -1 \\ 2 \end{bmatrix} \]
and
\[ v_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix}, \quad v_3 = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} \]

We can express the matrix \( A \) in terms of its action on the vectors \( v_1, v_2, v_3 \).

### Step 2: Form the Matrix from the Column Vectors
We can construct a matrix \( V \) from the column vectors \( v_1, v_2, v_3 \):
\[ V = \begin{bmatrix} 1 & 2 & 1 \\ 1 & 0 & 1 \\ 1 & 3 & -1 \end{bmatrix} \]

### Step 3: Form the Resulting Matrix
The resulting matrix from the transformations can be represented as:
\[ B = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 0 & -1 \\ 3 & 2 & 2 \end{bmatrix} \]

### Step 4: Set Up the Determinant Equation
We know that:
\[ A \cdot V = B \]
To find \( |A| \), we can use the property of determinants:
\[ |A| \cdot |V| = |B| \]

### Step 5: Calculate the Determinants
First, we need to calculate \( |V| \):
\[ |V| = \begin{vmatrix} 1 & 2 & 1 \\ 1 & 0 & 1 \\ 1 & 3 & -1 \end{vmatrix} \]
Calculating this determinant using cofactor expansion:
\[ = 1 \cdot \begin{vmatrix} 0 & 1 \\ 3 & -1 \end{vmatrix} - 2 \cdot \begin{vmatrix} 1 & 1 \\ 1 & -1 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 0 \\ 1 & 3 \end{vmatrix} \]
Calculating the 2x2 determinants:
\[ = 1 \cdot (0 \cdot -1 - 1 \cdot 3) - 2 \cdot (1 \cdot -1 - 1 \cdot 1) + 1 \cdot (1 \cdot 3 - 0 \cdot 1) \]
\[ = 1 \cdot (-3) - 2 \cdot (-2) + 1 \cdot 3 \]
\[ = -3 + 4 + 3 = 4 \]

Now calculate \( |B| \):
\[ |B| = \begin{vmatrix} 1 & -1 & 0 \\ 2 & 0 & -1 \\ 3 & 2 & 2 \end{vmatrix} \]
Calculating this determinant:
\[ = 1 \cdot \begin{vmatrix} 0 & -1 \\ 2 & 2 \end{vmatrix} + 1 \cdot \begin{vmatrix} 2 & -1 \\ 3 & 2 \end{vmatrix} \]
Calculating the 2x2 determinants:
\[ = 1 \cdot (0 \cdot 2 - (-1) \cdot 2) + 1 \cdot (2 \cdot 2 - (-1) \cdot 3) \]
\[ = 1 \cdot 2 + 1 \cdot (4 + 3) = 2 + 7 = 9 \]

### Step 6: Solve for \( |A| \)
Using the determinant property:
\[ |A| \cdot |V| = |B| \]
\[ |A| \cdot 4 = 9 \implies |A| = \frac{9}{4} \]

### Final Answer
Thus, the value of \( |A| \) is:
\[ \boxed{\frac{9}{4}} \]