Let `A` is `3 xx 3` matrix such that `Ax_1=B_1,Ax_2=B_2,Ax_3=B_3` where
`x_1=[[1],[1],[1]],x_2=[[0],[2],[1]],x_3=[[0],[0],[1]]` ,br> `B_1=[[1],[0],[0]],B_2=[[0],[2],[0]],B_3=[[0],[0],[2]]` then find `|A|`.

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the determinant of the matrix \( A \) given the equations \( Ax_1 = B_1 \), \( Ax_2 = B_2 \), and \( Ax_3 = B_3 \), we will follow these steps:

### Step 1: Define the Matrix \( A \)
Assume the matrix \( A \) is given by:
\[ A = \begin{pmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{pmatrix} \]

### Step 2: Set Up the Equations
Using the equations provided:
1. \( Ax_1 = B_1 \)
- \( A \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \)
- This gives us:
\[ \begin{aligned} a_1 + a_2 + a_3 &= 1 \quad (1) \\ a_4 + a_5 + a_6 &= 0 \quad (2) \\ a_7 + a_8 + a_9 &= 0 \quad (3) \end{aligned} \]

2. \( Ax_2 = B_2 \)
- \( A \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 2 \\ 0 \end{pmatrix} \)
- This gives us:
\[ \begin{aligned} 0 \cdot a_1 + 2a_2 + a_3 &= 0 \quad (4) \\ 0 \cdot a_4 + 2a_5 + a_6 &= 2 \quad (5) \\ 0 \cdot a_7 + 2a_8 + a_9 &= 0 \quad (6) \end{aligned} \]

3. \( Ax_3 = B_3 \)
- \( A \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 2 \end{pmatrix} \)
- This gives us:
\[ \begin{aligned} 0 \cdot a_1 + 0 \cdot a_2 + a_3 &= 0 \quad (7) \\ 0 \cdot a_4 + 0 \cdot a_5 + a_6 &= 0 \quad (8) \\ 0 \cdot a_7 + 0 \cdot a_8 + a_9 &= 2 \quad (9) \end{aligned} \]

### Step 3: Solve the Equations
From equation (7), we have:
\[ a_3 = 0 \]

From equation (8):
\[ a_6 = 0 \]

From equation (9):
\[ a_9 = 2 \]

Substituting \( a_3 = 0 \) into equation (1):
\[ a_1 + a_2 + 0 = 1 \implies a_1 + a_2 = 1 \quad (10) \]

Substituting \( a_6 = 0 \) into equation (5):
\[ 2a_5 + 0 = 2 \implies a_5 = 1 \]

Substituting \( a_5 = 1 \) into equation (2):
\[ a_4 + 1 + 0 = 0 \implies a_4 = -1 \]

Substituting \( a_9 = 2 \) into equation (3):
\[ a_7 + a_8 + 2 = 0 \implies a_7 + a_8 = -2 \quad (11) \]

Using equation (6) with \( a_9 = 2 \):
\[ 2a_8 + 2 = 0 \implies a_8 = -1 \]

Substituting \( a_8 = -1 \) into equation (11):
\[ a_7 - 1 = -2 \implies a_7 = -1 \]

### Step 4: Construct Matrix \( A \)
Now we have:
\[ \begin{aligned} a_1 &= 1 \\ a_2 &= 0 \\ a_3 &= 0 \\ a_4 &= -1 \\ a_5 &= 1 \\ a_6 &= 0 \\ a_7 &= -1 \\ a_8 &= -1 \\ a_9 &= 2 \end{aligned} \]
Thus, the matrix \( A \) is:
\[ A = \begin{pmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ -1 & -1 & 2 \end{pmatrix} \]

### Step 5: Calculate the Determinant of \( A \)
To find the determinant of \( A \):
\[ |A| = 1 \cdot \begin{vmatrix} 1 & 0 \\ -1 & 2 \end{vmatrix} - 0 + 0 \]
Calculating the 2x2 determinant:
\[ \begin{vmatrix} 1 & 0 \\ -1 & 2 \end{vmatrix} = (1)(2) - (0)(-1) = 2 \]
Thus,
\[ |A| = 1 \cdot 2 = 2 \]

### Final Answer
The determinant of matrix \( A \) is:
\[ \boxed{2} \]