Question

If the sum of the eigen values of the matrix of the quadratic form is zero, then the nature of the quadratic form is

            a. indefinite

             b. definite

             c. positive definite

            d. zero               ​

Answer 1

Answer:

I think the answer is C

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Answer 2

The nature of the quadratic form is Indefinite. Option A is the correct answer.

• Consider a matrix A, the eigenvalues of the matrix A can be determined using the formula |A – k I| = 0 where k is the eigenvalues of the matrix A.

• A matrix of order n can have a maximum of n eigenvalues.  

• Eigenvalues of a matrix can be 0, positive, and negative.

• If all the eigenvalues of the matrix are greater than 0, the quadratic form is considered as positive definite.

• If all the eigenvalues of the matrix are less than 0, the quadratic form is considered as negative definite.

• If all the eigenvalues of the matrix are less than or equal to zero, the quadratic form is considered as negative semi-definite.

• If all the eigenvalues of the matrix are greater than or equal to zero, the quadratic form is considered as positive semi-definite.

• If all the eigenvalues of the matrix have positive, negative, and zeroes included, the quadratic form is considered indefinite.

Therefore, Option A is the correct answer. The nature of the quadratic form is indefinite.