the value of the determinant
`|{:((a_(1)-b_(1))^(2),,(a_(1)-b_(2))^(2),,(a_(1)-b_(3))^(2),,(a_(1)-b_(4))^(2)),((a_(2)-b_(1))^(2),,(a_(2)-b_(2))^(2) ,,(a_(2)-b_(3))^(2),,(a_(2)-b_(4))^(2)),((a_(3)-b_(1))^(2),,(a_(3)-b_(2))^(2),,(a_(3)-b_(3))^(2),,(a_(3)-b_(4))^(2)),((a_(4)-b_(1))^(2),,(a_(4)-b_(2))^(2),,(a_(4)-b_(3))^(2),,(a_(4)-b_(4))^(2)):}|` is

Options

A. dependant on `a_(i),i=1,2,3,4`
B. dependant on `b_(i),i=1,2,3,4`
C. dependant on `a_(ij), b_(i) i= 1,2,3,4`
D. 0

Explanation

To find the value of the determinant

\[ D = \begin{vmatrix} (a_1 - b_1)^2 & (a_1 - b_2)^2 & (a_1 - b_3)^2 & (a_1 - b_4)^2 \\ (a_2 - b_1)^2 & (a_2 - b_2)^2 & (a_2 - b_3)^2 & (a_2 - b_4)^2 \\ (a_3 - b_1)^2 & (a_3 - b_2)^2 & (a_3 - b_3)^2 & (a_3 - b_4)^2 \\ (a_4 - b_1)^2 & (a_4 - b_2)^2 & (a_4 - b_3)^2 & (a_4 - b_4)^2 \end{vmatrix} \]

we will simplify the determinant using the identity \( (x - y)^2 = x^2 + y^2 - 2xy \).

### Step 1: Expand each element using the identity

Using the identity \( (x - y)^2 = x^2 + y^2 - 2xy \), we can rewrite each element of the determinant:

- The first row becomes:
- \((a_1 - b_1)^2 = a_1^2 + b_1^2 - 2a_1b_1\)
- \((a_1 - b_2)^2 = a_1^2 + b_2^2 - 2a_1b_2\)
- \((a_1 - b_3)^2 = a_1^2 + b_3^2 - 2a_1b_3\)
- \((a_1 - b_4)^2 = a_1^2 + b_4^2 - 2a_1b_4\)

- The second row becomes:
- \((a_2 - b_1)^2 = a_2^2 + b_1^2 - 2a_2b_1\)
- \((a_2 - b_2)^2 = a_2^2 + b_2^2 - 2a_2b_2\)
- \((a_2 - b_3)^2 = a_2^2 + b_3^2 - 2a_2b_3\)
- \((a_2 - b_4)^2 = a_2^2 + b_4^2 - 2a_2b_4\)

- The third and fourth rows are similarly expanded.

### Step 2: Rewrite the determinant

Now, the determinant can be expressed as:

\[ D = \begin{vmatrix} a_1^2 + b_1^2 - 2a_1b_1 & a_1^2 + b_2^2 - 2a_1b_2 & a_1^2 + b_3^2 - 2a_1b_3 & a_1^2 + b_4^2 - 2a_1b_4 \\ a_2^2 + b_1^2 - 2a_2b_1 & a_2^2 + b_2^2 - 2a_2b_2 & a_2^2 + b_3^2 - 2a_2b_3 & a_2^2 + b_4^2 - 2a_2b_4 \\ a_3^2 + b_1^2 - 2a_3b_1 & a_3^2 + b_2^2 - 2a_3b_2 & a_3^2 + b_3^2 - 2a_3b_3 & a_3^2 + b_4^2 - 2a_3b_4 \\ a_4^2 + b_1^2 - 2a_4b_1 & a_4^2 + b_2^2 - 2a_4b_2 & a_4^2 + b_3^2 - 2a_4b_3 & a_4^2 + b_4^2 - 2a_4b_4 \end{vmatrix} \]

### Step 3: Factor out common terms

Notice that each row can be expressed as a linear combination of the squares of \(a_i\) and \(b_j\) and their products. This suggests that there may be dependencies among the rows.

### Step 4: Check for linear dependence

If we observe the structure of the determinant, we can see that each row can be expressed in terms of the others, indicating that the rows are linearly dependent.

### Step 5: Conclusion

Since the rows of the determinant are linearly dependent, the value of the determinant is:

\[ D = 0 \]