The given quadratic form is (x₁ - x₂)² + 2x₃². To find the matrix A corresponding to this quadratic form, we need to identify the coefficients of the quadratic terms.

First, let's expand the given quadratic form:

(x₁ - x₂)² + 2x₃² = (x₁ - x₂)(x₁ - x₂) + 2x₃²
= x₁² - 2x₁x₂ + x₂² + 2x₃²

We can see that the coefficient of x₁² is 1, the coefficient of x₂² is 1, and the coefficient of x₃² is 2. The coefficient of x₁x₂ is -2.

Now, we can construct the matrix A using these coefficients:

A = |1 -2 0|
|-2 1 0|
|0 0 2|

The trace of a matrix is the sum of the elements on its main diagonal. In this case, the main diagonal elements of A are 1, 1, and 2.

Therefore, the trace of matrix A is:

trace(A) = 1 + 1 + 2
= 4

Hence, the trace of matrix A is 4.

In summary:
- The given quadratic form is (x₁ - x₂)² + 2x₃².
- The matrix A corresponding to this quadratic form is:
A = |1 -2 0|
|-2 1 0|
|0 0 2|
- The trace of matrix A is 4, which is the sum of the main diagonal elements.