Introduction:
In linear algebra, a quadratic form is a homogeneous polynomial of degree two in a number of variables. It is useful in many areas of mathematics, including matrix theory, differential geometry, and physics. In this question, we are given a quadratic form Q=4x1x2 – 5x22′ and we need to find the real symmetric matrix C corresponding to it. In this answer, we will explain how to find the matrix C step by step.

Step 1: Write the Quadratic Form in Matrix Notation:
The given quadratic form Q can be written in matrix notation as follows:

Q = [x1 x2] [0 2] [4 -5] [x1 x2]T

Here, [0 2] and [4 -5] are the coefficient matrices of x1x2 and x22′ respectively. We can combine these two matrices to get the real symmetric matrix C.

Step 2: Combine the Coefficient Matrices:
To get the matrix C, we need to combine the coefficient matrices [0 2] and [4 -5] as follows:

C = [0 2] [4 -5] = [0 4] [2 -5]

Step 3: Verify that C is a Real Symmetric Matrix:
To verify that C is a real symmetric matrix, we need to check two conditions: (i) C is a square matrix, and (ii) C is equal to its transpose.

(i) C is a 2x2 square matrix, so the first condition is satisfied.

(ii) The transpose of C is given by:

CT = [0 4] [2 -5]T = [0 2] [4 -5]

Since C is equal to its transpose, the second condition is also satisfied.

Therefore, C is a real symmetric matrix.

Conclusion:
In this answer, we have shown how to find the real symmetric matrix C corresponding to the given quadratic form Q=4x1x2 – 5x22′. We have used matrix notation to express the quadratic form, combined the coefficient matrices to get the matrix C, and verified that C is a real symmetric matrix.