If z is a complex number and all `a_i` 's and `b_i` 's are real numbers, then
`Delta=|(a_1z+b_1barz,a_2z+b_2barz,a_3z+b_3barz),(b_1z+a_1barz,b_2z+a_2barz,b_3z+a_3barz),(b_1z+a_1,b_2z+a_2,b_3z+a_3)|=`

Options

A. `(a_1a_2a_3+b_1b_2b_3)^2|z|^2`
B. 0
C. `(a_1a_2a_3-b_1b_2b_3)^2|z|^2`
D. none

Answer

To solve the determinant given in the question, we will follow a systematic approach. The determinant is defined as:

\[ \Delta = \begin{vmatrix} a_1 z + b_1 \bar{z} & a_2 z + b_2 \bar{z} & a_3 z + b_3 \bar{z} \\ b_1 z + a_1 \bar{z} & b_2 z + a_2 \bar{z} & b_3 z + a_3 \bar{z} \\ b_1 z + a_1 & b_2 z + a_2 & b_3 z + a_3 \end{vmatrix} \]

### Step 1: Write the Determinant
We begin by rewriting the determinant as given:

\[ \Delta = \begin{vmatrix} a_1 z + b_1 \bar{z} & a_2 z + b_2 \bar{z} & a_3 z + b_3 \bar{z} \\ b_1 z + a_1 \bar{z} & b_2 z + a_2 \bar{z} & b_3 z + a_3 \bar{z} \\ b_1 z + a_1 & b_2 z + a_2 & b_3 z + a_3 \end{vmatrix} \]

### Step 2: Analyze the Rows
Notice that the first two rows contain terms involving \(z\) and \(\bar{z}\), while the third row does not contain any \(\bar{z}\) terms. This suggests that we can manipulate the determinant to simplify our calculations.

### Step 3: Row Operations
We can perform row operations to simplify the determinant. Specifically, we can subtract the third row from the first and second rows:

\[ R_1 \rightarrow R_1 - R_3 \]
\[ R_2 \rightarrow R_2 - R_3 \]

This gives us:

\[ \Delta = \begin{vmatrix} (a_1 z + b_1 \bar{z} - (b_1 z + a_1)) & (a_2 z + b_2 \bar{z} - (b_2 z + a_2)) & (a_3 z + b_3 \bar{z} - (b_3 z + a_3)) \\ (b_1 z + a_1 \bar{z} - (b_1 z + a_1)) & (b_2 z + a_2 \bar{z} - (b_2 z + a_2)) & (b_3 z + a_3 \bar{z} - (b_3 z + a_3)) \\ b_1 z + a_1 & b_2 z + a_2 & b_3 z + a_3 \end{vmatrix} \]

### Step 4: Simplifying the Rows
After performing the row operations, we simplify the determinant:

\[ \Delta = \begin{vmatrix} (b_1 \bar{z} - a_1) & (b_2 \bar{z} - a_2) & (b_3 \bar{z} - a_3) \\ (a_1 \bar{z} - a_1) & (a_2 \bar{z} - a_2) & (a_3 \bar{z} - a_3) \\ b_1 z + a_1 & b_2 z + a_2 & b_3 z + a_3 \end{vmatrix} \]

### Step 5: Recognizing Zero Rows
Notice that if we set \(z = 0\), the first two rows will become zero, leading to:

\[ \Delta = 0 \]

### Conclusion
Thus, the value of the determinant \(\Delta\) is:

\[ \Delta = 0 \]