Linear AlgebraSolutionA symmetric n n real matrix A is said

Linear Algebra:

Solution

A symmetric n × n real matrix A is said to be positive definite if the scalar x^TAx is positive for every non-zero column vector x of  real numbers.

We have x^TAx = 3x₁² - x₁ x₂ + x₂² = [ ( x₁3)² - 2* ( x₁ 3)( x₂ /23) + ( x₂ / 23)²] + ( x₂² - 3/4x₂²) = ( x₁ -x₂ 3/2)² + x₂²/4 . Now, we know that the square of any real number, whether negative or positive, is positive. Therefore, x^TAx = 3x₁² - x₁ x₂ + x₂² = ( x₁ -x₂ 3/2)² + x₂²/4 > 0.However, since A is not a symmetric matrix, it is not a positive definite matrix.