If `|{:(a,b,0),(0,a,b),(b,0,a):}|` = 0, `(a ne 0 )` then

Options

A. a is one of cube root of unity
B. b is one of cube root of unity
C. (a/b) is one of cube root of unity
D. (a/b) is one of cube root of -1

Answer

To solve the determinant equation given by:

\[ \begin{vmatrix} a & b & 0 \\ 0 & a & b \\ b & 0 & a \end{vmatrix} = 0 \]

we can follow these steps:

### Step 1: Calculate the Determinant
We will calculate the determinant using the formula for a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + 0(dh - eg) \]

For our matrix:

\[ \begin{vmatrix} a & b & 0 \\ 0 & a & b \\ b & 0 & a \end{vmatrix} \]

We can expand it as follows:

\[ = a \begin{vmatrix} a & b \\ 0 & a \end{vmatrix} - b \begin{vmatrix} 0 & b \\ b & a \end{vmatrix} + 0 \]

### Step 2: Calculate the 2x2 Determinants
Now we calculate the two 2x2 determinants:

1. For the first determinant:

\[ \begin{vmatrix} a & b \\ 0 & a \end{vmatrix} = a \cdot a - b \cdot 0 = a^2 \]

2. For the second determinant:

\[ \begin{vmatrix} 0 & b \\ b & a \end{vmatrix} = 0 \cdot a - b \cdot b = -b^2 \]

### Step 3: Substitute Back into the Determinant
Substituting these back into our determinant calculation:

\[ = a(a^2) - b(-b^2) = a^3 + b^3 \]

### Step 4: Set the Determinant to Zero
We are given that this determinant equals zero:

\[ a^3 + b^3 = 0 \]

### Step 5: Factor the Equation
We can factor this equation using the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):

\[ (a + b)(a^2 - ab + b^2) = 0 \]

### Step 6: Analyze the Factors
Since \( a \neq 0 \) (given), we can conclude that:

1. \( a + b = 0 \) or
2. \( a^2 - ab + b^2 = 0 \)

From \( a + b = 0 \), we have:

\[ b = -a \]

### Step 7: Find the Ratio \( \frac{a}{b} \)
Substituting \( b = -a \):

\[ \frac{a}{b} = \frac{a}{-a} = -1 \]

### Conclusion
Thus, we conclude that:

\[ \frac{a}{b} = -1 \]

This means \( \frac{a}{b} \) is a cube root of \(-1\). Therefore, the correct option is:

**Option D: \( \frac{a}{b} \) is a cube root of -1.**